WEBVTT 00:00:00.000 --> 00:00:02.400 align:middle line:90% [SQUEAKING] 00:00:02.400 --> 00:00:04.320 align:middle line:90% [RUSTLING] 00:00:04.320 --> 00:00:05.760 align:middle line:90% [CLICKING] 00:00:05.760 --> 00:00:10.560 align:middle line:90% 00:00:10.560 --> 00:00:12.280 align:middle line:84% TOBIAS COLDING: Last time, we started 00:00:12.280 --> 00:00:15.300 align:middle line:84% looking at what's called the Riemann integrals. 00:00:15.300 --> 00:00:24.760 align:middle line:90% 00:00:24.760 --> 00:00:27.960 align:middle line:84% For the Riemann integrals, we have a function, 00:00:27.960 --> 00:00:34.920 align:middle line:84% f, that is defined on some interval from a to b into R. 00:00:34.920 --> 00:00:36.910 align:middle line:90% And this function is bounded. 00:00:36.910 --> 00:00:40.160 align:middle line:90% 00:00:40.160 --> 00:00:50.390 align:middle line:84% And then we have P here is a partition of this interval. 00:00:50.390 --> 00:00:53.520 align:middle line:90% 00:00:53.520 --> 00:00:58.880 align:middle line:84% And so this means that we have a number of dividing points, 00:00:58.880 --> 00:01:06.900 align:middle line:84% where a is the first, and then we have x1, x0. 00:01:06.900 --> 00:01:14.860 align:middle line:84% That's a, x1, x2, and so on, up to, say, xn. 00:01:14.860 --> 00:01:22.420 align:middle line:84% And that is the final point, end point, of the interval. 00:01:22.420 --> 00:01:24.460 align:middle line:90% So this here is a partition. 00:01:24.460 --> 00:01:26.090 align:middle line:90% These are the dividing points. 00:01:26.090 --> 00:01:33.140 align:middle line:90% 00:01:33.140 --> 00:01:38.160 align:middle line:84% Then we look at the length of each of these subintervals. 00:01:38.160 --> 00:01:44.620 align:middle line:84% So we have the subintervals from xi minus 1 to xi. 00:01:44.620 --> 00:01:49.140 align:middle line:84% And the length of one of these subintervals 00:01:49.140 --> 00:01:52.260 align:middle line:90% we denote by delta xi. 00:01:52.260 --> 00:01:56.000 align:middle line:84% And so this is xi minus xi minus 1. 00:01:56.000 --> 00:01:59.360 align:middle line:90% 00:01:59.360 --> 00:02:03.660 align:middle line:90% And then we look at-- 00:02:03.660 --> 00:02:06.740 align:middle line:84% so here, if you have the function here, 00:02:06.740 --> 00:02:08.100 align:middle line:90% this is the function f. 00:02:08.100 --> 00:02:12.240 align:middle line:90% 00:02:12.240 --> 00:02:17.440 align:middle line:84% And here, we have the integral from a to b. 00:02:17.440 --> 00:02:20.460 align:middle line:84% And so this here is the first dividing point. 00:02:20.460 --> 00:02:22.360 align:middle line:90% This is the last dividing point. 00:02:22.360 --> 00:02:26.750 align:middle line:84% And then we have these other dividing points in between. 00:02:26.750 --> 00:02:29.960 align:middle line:90% 00:02:29.960 --> 00:02:32.880 align:middle line:90% And this is maybe like that. 00:02:32.880 --> 00:02:38.880 align:middle line:84% Then we look at the function on one of these subintervals. 00:02:38.880 --> 00:02:39.970 align:middle line:90% So we look at-- 00:02:39.970 --> 00:02:45.400 align:middle line:90% 00:02:45.400 --> 00:02:47.120 align:middle line:90% we look at the function. 00:02:47.120 --> 00:02:50.900 align:middle line:84% And we restrict it to one of these subintervals. 00:02:50.900 --> 00:02:56.160 align:middle line:90% 00:02:56.160 --> 00:02:59.440 align:middle line:84% And then remember the function is bounded. 00:02:59.440 --> 00:03:02.140 align:middle line:84% And so we're looking at it on one of these. 00:03:02.140 --> 00:03:06.340 align:middle line:84% And we defined capital Mi to be the sup 00:03:06.340 --> 00:03:09.400 align:middle line:90% of f over this subinterval. 00:03:09.400 --> 00:03:14.000 align:middle line:90% 00:03:14.000 --> 00:03:17.920 align:middle line:84% Likewise, we defined little mi to be the inf. 00:03:17.920 --> 00:03:22.540 align:middle line:90% 00:03:22.540 --> 00:03:25.900 align:middle line:84% And this makes sense, the finite number, 00:03:25.900 --> 00:03:27.390 align:middle line:90% because the function is bounded. 00:03:27.390 --> 00:03:30.060 align:middle line:90% 00:03:30.060 --> 00:03:34.080 align:middle line:84% And then we look at the upper and the lower sum. 00:03:34.080 --> 00:03:39.780 align:middle line:90% So the upper sum. 00:03:39.780 --> 00:03:44.180 align:middle line:84% So the upper sum is that we have this given partition. 00:03:44.180 --> 00:03:52.740 align:middle line:84% And we look at the sum here of the Mi times delta xi. 00:03:52.740 --> 00:03:54.100 align:middle line:90% And we're summing. 00:03:54.100 --> 00:03:56.400 align:middle line:90% This is from 1 to n. 00:03:56.400 --> 00:04:03.440 align:middle line:84% So this is the upper sum of the function for this partition. 00:04:03.440 --> 00:04:05.790 align:middle line:84% And likewise, we look at the lower sum. 00:04:05.790 --> 00:04:10.600 align:middle line:90% 00:04:10.600 --> 00:04:13.200 align:middle line:84% This is where we sum the infimum. 00:04:13.200 --> 00:04:17.120 align:middle line:84% And we weight them by the length of the intervals. 00:04:17.120 --> 00:04:24.200 align:middle line:84% So we look at mi and then delta xi and i equal to 1 to n. 00:04:24.200 --> 00:04:29.780 align:middle line:84% And this is the lower sum for this partition. 00:04:29.780 --> 00:04:33.280 align:middle line:90% 00:04:33.280 --> 00:04:35.250 align:middle line:90% And then just using-- 00:04:35.250 --> 00:04:38.640 align:middle line:90% 00:04:38.640 --> 00:04:41.480 align:middle line:90% then just using that-- 00:04:41.480 --> 00:04:46.560 align:middle line:84% so of course, for these mi, like these here, 00:04:46.560 --> 00:04:49.760 align:middle line:84% it's the inf and sum on the same interval. 00:04:49.760 --> 00:04:54.700 align:middle line:84% So little mi is less than or equal to capital Mi. 00:04:54.700 --> 00:04:59.040 align:middle line:84% But this also means that these weights are the same. 00:04:59.040 --> 00:05:01.140 align:middle line:84% If this here is less than that, it 00:05:01.140 --> 00:05:07.080 align:middle line:84% means that the lower sum is less than or equal to the upper sum. 00:05:07.080 --> 00:05:11.100 align:middle line:90% 00:05:11.100 --> 00:05:14.260 align:middle line:84% Now, the second thing we looked at is-- 00:05:14.260 --> 00:05:16.880 align:middle line:90% so again, we looked at-- 00:05:16.880 --> 00:05:19.780 align:middle line:84% so again, we have this partition P. 00:05:19.780 --> 00:05:23.600 align:middle line:84% And we looked at P1 as a subpartition-- 00:05:23.600 --> 00:05:31.540 align:middle line:90% 00:05:31.540 --> 00:05:34.320 align:middle line:90% subpartition or a refinement. 00:05:34.320 --> 00:05:38.340 align:middle line:90% 00:05:38.340 --> 00:05:58.000 align:middle line:84% This is if P1 has the same dividing points as P 00:05:58.000 --> 00:05:59.820 align:middle line:90% and then presumably some-- 00:05:59.820 --> 00:06:03.280 align:middle line:90% 00:06:03.280 --> 00:06:06.380 align:middle line:90% then-- I mean, then some more. 00:06:06.380 --> 00:06:09.800 align:middle line:90% 00:06:09.800 --> 00:06:12.040 align:middle line:84% But I mean, you could have that-- you 00:06:12.040 --> 00:06:15.700 align:middle line:84% could have that the true partition was exactly the same. 00:06:15.700 --> 00:06:19.840 align:middle line:84% And you would still say that one is a trivial subpartition 00:06:19.840 --> 00:06:22.200 align:middle line:90% of the other. 00:06:22.200 --> 00:06:27.260 align:middle line:84% But in general, a subpartition would have more dividing points. 00:06:27.260 --> 00:06:33.520 align:middle line:84% And so then the thing is that if you're looking at-- so let's say 00:06:33.520 --> 00:06:35.560 align:middle line:90% that-- 00:06:35.560 --> 00:06:41.960 align:middle line:84% so that P here, again, has the partitions. 00:06:41.960 --> 00:06:50.700 align:middle line:84% This is the partition that comes from P. And let's 00:06:50.700 --> 00:06:53.540 align:middle line:90% say that P1 here-- 00:06:53.540 --> 00:06:57.460 align:middle line:84% let me illustrate it as if P1 had one additional. 00:06:57.460 --> 00:07:01.020 align:middle line:84% So here, you have x0, which is a. 00:07:01.020 --> 00:07:04.060 align:middle line:90% Here, you have xn, which is b. 00:07:04.060 --> 00:07:06.860 align:middle line:90% And then you have two other. 00:07:06.860 --> 00:07:09.380 align:middle line:90% Say this is x1 maybe. 00:07:09.380 --> 00:07:11.900 align:middle line:90% And this here is xk. 00:07:11.900 --> 00:07:13.520 align:middle line:90% Then you have a bunch of other. 00:07:13.520 --> 00:07:15.500 align:middle line:90% This is the partition p. 00:07:15.500 --> 00:07:25.480 align:middle line:84% And then p1, let's say, have one more dividing point. 00:07:25.480 --> 00:07:27.500 align:middle line:84% It could have many more dividing points. 00:07:27.500 --> 00:07:29.520 align:middle line:90% But if you're looking at-- 00:07:29.520 --> 00:07:34.980 align:middle line:90% so you have here xi minus 1. 00:07:34.980 --> 00:07:43.300 align:middle line:84% And then you have xi and then maybe the P1, this partition, 00:07:43.300 --> 00:07:48.444 align:middle line:84% have an additional dividing point, which is called that one. 00:07:48.444 --> 00:07:53.560 align:middle line:84% Well, then the contribution that you get from-- 00:07:53.560 --> 00:07:56.960 align:middle line:90% to the lower and upper sum-- 00:07:56.960 --> 00:08:00.280 align:middle line:84% so the contribution that you get from this 00:08:00.280 --> 00:08:07.120 align:middle line:84% or that for P, you can think about that as mi. 00:08:07.120 --> 00:08:10.800 align:middle line:84% This is the contribution from the lowest sum. 00:08:10.800 --> 00:08:14.580 align:middle line:84% And here, you have the contribution from the upper sum. 00:08:14.580 --> 00:08:17.880 align:middle line:90% 00:08:17.880 --> 00:08:28.080 align:middle line:84% But now you see that for the subpartition, then 00:08:28.080 --> 00:08:29.380 align:middle line:90% you have that-- 00:08:29.380 --> 00:08:32.100 align:middle line:84% so the intervals for the subpartition-- 00:08:32.100 --> 00:08:36.679 align:middle line:84% this here is the interval for the original partition P. 00:08:36.679 --> 00:08:41.080 align:middle line:84% But for the subpartition, this is now 00:08:41.080 --> 00:08:46.260 align:middle line:84% divided into two intervals, this here and that here. 00:08:46.260 --> 00:08:50.220 align:middle line:90% And so you have that. 00:08:50.220 --> 00:08:55.300 align:middle line:84% So this thing here, this is, in this case, split into two. 00:08:55.300 --> 00:08:59.620 align:middle line:84% And one of them is y minus xi minus 1. 00:08:59.620 --> 00:09:02.700 align:middle line:90% And the other is xi minus 1. 00:09:02.700 --> 00:09:08.300 align:middle line:90% 00:09:08.300 --> 00:09:10.980 align:middle line:90% If you're looking at the-- 00:09:10.980 --> 00:09:18.300 align:middle line:84% say, the lower sum for this subpartition, 00:09:18.300 --> 00:09:21.540 align:middle line:84% so this thing here, so the contribution 00:09:21.540 --> 00:09:25.180 align:middle line:84% that you get from this here, where you have-- now 00:09:25.180 --> 00:09:27.740 align:middle line:90% you have two intervals. 00:09:27.740 --> 00:09:29.900 align:middle line:90% And so you're getting a-- 00:09:29.900 --> 00:09:36.740 align:middle line:84% you're getting a contribution, which is the inf on this thing 00:09:36.740 --> 00:09:40.180 align:middle line:84% here, on this little interval here 00:09:40.180 --> 00:09:46.960 align:middle line:90% of f, times y minus xi minus 1. 00:09:46.960 --> 00:09:49.060 align:middle line:84% And then you get a second contribution, 00:09:49.060 --> 00:10:00.440 align:middle line:84% which is inf on this interval here, from y to xi of f. 00:10:00.440 --> 00:10:04.480 align:middle line:90% And then it's xi minus y. 00:10:04.480 --> 00:10:12.480 align:middle line:84% So this is the contribution that you get from this interval here. 00:10:12.480 --> 00:10:17.840 align:middle line:84% But this thing here is smaller or equal to the-- 00:10:17.840 --> 00:10:18.840 align:middle line:90% it's actually-- sorry. 00:10:18.840 --> 00:10:22.240 align:middle line:84% This here is larger or equal to the inf 00:10:22.240 --> 00:10:25.760 align:middle line:84% where you take the inf over the larger interval that goes 00:10:25.760 --> 00:10:28.140 align:middle line:90% from this point to that point. 00:10:28.140 --> 00:10:34.320 align:middle line:84% So you have that this is larger than y minus xi minus 1. 00:10:34.320 --> 00:10:39.760 align:middle line:84% This thing here is also larger than this little mi. 00:10:39.760 --> 00:10:44.260 align:middle line:90% And then it's xi minus y. 00:10:44.260 --> 00:10:47.660 align:middle line:84% And now you can just factor out-- so you're getting mi. 00:10:47.660 --> 00:10:50.740 align:middle line:84% And then when you add these two together, 00:10:50.740 --> 00:10:56.020 align:middle line:84% then you see that this gives you exactly xi minus xi minus 1. 00:10:56.020 --> 00:11:01.340 align:middle line:84% So this is little mi times delta xi. 00:11:01.340 --> 00:11:03.380 align:middle line:90% So this means that if you-- 00:11:03.380 --> 00:11:05.700 align:middle line:84% so you see that the contribution that you 00:11:05.700 --> 00:11:09.220 align:middle line:84% get on this little interval for this 00:11:09.220 --> 00:11:14.980 align:middle line:84% is larger than the contribution that you get from this-- 00:11:14.980 --> 00:11:17.280 align:middle line:90% larger, equal to. 00:11:17.280 --> 00:11:21.100 align:middle line:84% And so when you're summing over all intervals, 00:11:21.100 --> 00:11:34.500 align:middle line:84% then you get that if P1 is a refinement or subpartition, then 00:11:34.500 --> 00:11:39.640 align:middle line:84% this lower here, the lower sum, go up. 00:11:39.640 --> 00:11:42.920 align:middle line:90% 00:11:42.920 --> 00:11:45.200 align:middle line:90% So this lower sum go up. 00:11:45.200 --> 00:11:50.520 align:middle line:84% And if you do it now, if you do the same stuff 00:11:50.520 --> 00:11:53.080 align:middle line:84% for the upper sum, then this here 00:11:53.080 --> 00:11:57.240 align:middle line:84% would-- instead of being inf, it would have to be the sup. 00:11:57.240 --> 00:12:01.160 align:middle line:84% This here would have to be the sup. 00:12:01.160 --> 00:12:03.880 align:middle line:84% But the sup on the smaller interval 00:12:03.880 --> 00:12:07.720 align:middle line:84% is now then the-- this inequality would reverse. 00:12:07.720 --> 00:12:09.600 align:middle line:90% It's less than or equal to. 00:12:09.600 --> 00:12:12.760 align:middle line:84% And then it would come to sup on the larger interval. 00:12:12.760 --> 00:12:14.840 align:middle line:90% This here remains the same. 00:12:14.840 --> 00:12:18.200 align:middle line:84% This here again becomes a larger one. 00:12:18.200 --> 00:12:22.180 align:middle line:84% So this here becomes less than this. 00:12:22.180 --> 00:12:25.520 align:middle line:84% This here becomes less than this. 00:12:25.520 --> 00:12:30.980 align:middle line:84% And so what you see is that the sup here go down. 00:12:30.980 --> 00:12:36.360 align:middle line:90% So the lower sum go up. 00:12:36.360 --> 00:12:46.580 align:middle line:84% And if you're looking at the upper sum, this here, decrease. 00:12:46.580 --> 00:12:48.940 align:middle line:90% So you have that. 00:12:48.940 --> 00:12:54.220 align:middle line:84% And so putting this together, you 00:12:54.220 --> 00:12:59.300 align:middle line:84% have that the lower sum for the partition P 00:12:59.300 --> 00:13:05.100 align:middle line:84% is less than the lower sum for this subpartition, which 00:13:05.100 --> 00:13:11.380 align:middle line:84% is less than the upper sum for the subpartition, which is less 00:13:11.380 --> 00:13:17.140 align:middle line:90% than the original upper sum. 00:13:17.140 --> 00:13:20.780 align:middle line:90% So you have that. 00:13:20.780 --> 00:13:24.460 align:middle line:90% And so this is like a key fact. 00:13:24.460 --> 00:13:28.130 align:middle line:84% It's an obvious fact, but it's a key fact. 00:13:28.130 --> 00:13:32.140 align:middle line:90% 00:13:32.140 --> 00:13:39.110 align:middle line:84% So that was the upper and lower sum. 00:13:39.110 --> 00:13:47.510 align:middle line:84% So now we define the upper and lower integrals. 00:13:47.510 --> 00:13:59.130 align:middle line:84% So we define the upper and lower Riemann integrals. 00:13:59.130 --> 00:14:07.390 align:middle line:90% 00:14:07.390 --> 00:14:14.390 align:middle line:84% You define them as follows, that the upper one 00:14:14.390 --> 00:14:15.580 align:middle line:90% we denote like this. 00:14:15.580 --> 00:14:18.190 align:middle line:90% 00:14:18.190 --> 00:14:26.010 align:middle line:84% And this here is now the inf of all partitions of the upper sum. 00:14:26.010 --> 00:14:30.030 align:middle line:90% 00:14:30.030 --> 00:14:34.390 align:middle line:84% And likewise, we define the lower Riemann integral 00:14:34.390 --> 00:14:39.090 align:middle line:84% of this function, this bounded function, 00:14:39.090 --> 00:14:47.250 align:middle line:84% to be the sup of the lower sums, like that. 00:14:47.250 --> 00:14:49.690 align:middle line:90% So sup of all partitions. 00:14:49.690 --> 00:14:52.690 align:middle line:84% So that is the upper and lower sums-- 00:14:52.690 --> 00:14:55.210 align:middle line:90% Riemann integrals. 00:14:55.210 --> 00:15:02.770 align:middle line:84% And so now we say that the function, this bounded function 00:15:02.770 --> 00:15:05.830 align:middle line:90% f, is Riemann integrable. 00:15:05.830 --> 00:15:14.450 align:middle line:90% 00:15:14.450 --> 00:15:26.350 align:middle line:84% So f here-- so f is said to be a Riemann integrable. 00:15:26.350 --> 00:15:34.050 align:middle line:90% 00:15:34.050 --> 00:15:41.470 align:middle line:84% And we write that f here is like this. 00:15:41.470 --> 00:15:45.650 align:middle line:84% So it means that it's Riemann integrable. 00:15:45.650 --> 00:15:56.898 align:middle line:84% If the following is the case, so if this lower integral here 00:15:56.898 --> 00:16:04.970 align:middle line:84% is equal to the upper integral, and if this is the case, 00:16:04.970 --> 00:16:06.510 align:middle line:90% then we just write-- 00:16:06.510 --> 00:16:23.590 align:middle line:84% and then the Riemann integral of f is then denoted by-- 00:16:23.590 --> 00:16:25.510 align:middle line:90% just like that. 00:16:25.510 --> 00:16:30.350 align:middle line:90% 00:16:30.350 --> 00:16:34.850 align:middle line:84% And so this is so this is Riemann's way 00:16:34.850 --> 00:16:36.890 align:middle line:90% of defining integrals. 00:16:36.890 --> 00:16:39.690 align:middle line:84% There are other ways, but this is-- 00:16:39.690 --> 00:16:49.890 align:middle line:84% I think this was the first way where you have a definition that 00:16:49.890 --> 00:16:50.740 align:middle line:90% really made sense. 00:16:50.740 --> 00:16:54.130 align:middle line:90% 00:16:54.130 --> 00:16:59.890 align:middle line:84% So now the first thing I want to observe 00:16:59.890 --> 00:17:07.650 align:middle line:84% is that for a general function, you always have-- 00:17:07.650 --> 00:17:16.609 align:middle line:84% so in general, so we always have-- whether the function is 00:17:16.609 --> 00:17:19.290 align:middle line:84% Riemann integrable or not, we always 00:17:19.290 --> 00:17:26.290 align:middle line:84% have that the lower integral here, the lower Riemann integral 00:17:26.290 --> 00:17:28.710 align:middle line:84% is less than or equal to the upper. 00:17:28.710 --> 00:17:33.990 align:middle line:90% 00:17:33.990 --> 00:17:36.790 align:middle line:90% And so why is that the case? 00:17:36.790 --> 00:17:38.180 align:middle line:90% So this is because-- 00:17:38.180 --> 00:17:44.330 align:middle line:90% 00:17:44.330 --> 00:17:44.830 align:middle line:90% why? 00:17:44.830 --> 00:17:48.730 align:middle line:84% This is because if you take a partition-- 00:17:48.730 --> 00:17:54.870 align:middle line:90% 00:17:54.870 --> 00:17:56.690 align:middle line:90% so fix a partition. 00:17:56.690 --> 00:18:04.710 align:middle line:90% 00:18:04.710 --> 00:18:12.830 align:middle line:84% Then you have that the lower sum here 00:18:12.830 --> 00:18:19.630 align:middle line:84% is, of course, less than or equal to this thing 00:18:19.630 --> 00:18:28.430 align:middle line:84% here, because this here was the sup of all partitions. 00:18:28.430 --> 00:18:31.570 align:middle line:90% So you have that. 00:18:31.570 --> 00:18:35.190 align:middle line:84% But in fact, what you also have, that you-- 00:18:35.190 --> 00:18:52.810 align:middle line:84% but more importantly, you have that this L of f here, 00:18:52.810 --> 00:19:08.450 align:middle line:84% I claim that this is a less than or equal to the upper sum 00:19:08.450 --> 00:19:16.310 align:middle line:84% for any other partition, for any partition. 00:19:16.310 --> 00:19:21.490 align:middle line:90% 00:19:21.490 --> 00:19:23.550 align:middle line:84% So I claim that you always have this here. 00:19:23.550 --> 00:19:27.370 align:middle line:84% If you take two partition P and P star, 00:19:27.370 --> 00:19:29.790 align:middle line:84% then the lower sum for one is less than 00:19:29.790 --> 00:19:31.930 align:middle line:84% or equal to the upper sum for the other. 00:19:31.930 --> 00:19:33.550 align:middle line:90% Why is that the case? 00:19:33.550 --> 00:19:35.810 align:middle line:84% This just uses that inequality over there. 00:19:35.810 --> 00:19:39.150 align:middle line:90% 00:19:39.150 --> 00:19:42.890 align:middle line:84% It uses that together with this inequality. 00:19:42.890 --> 00:19:46.150 align:middle line:90% So let's see. 00:19:46.150 --> 00:19:47.360 align:middle line:90% So if you take-- 00:19:47.360 --> 00:19:51.950 align:middle line:90% 00:19:51.950 --> 00:20:02.830 align:middle line:84% if P here is a partition and P star is a partition, 00:20:02.830 --> 00:20:05.830 align:middle line:84% then you could take a partition-- 00:20:05.830 --> 00:20:16.310 align:middle line:84% so let P bar be the partition that 00:20:16.310 --> 00:20:41.460 align:middle line:84% has all the dividing points of P and all the dividing points of P 00:20:41.460 --> 00:20:41.960 align:middle line:90% star. 00:20:41.960 --> 00:20:44.610 align:middle line:90% 00:20:44.610 --> 00:20:47.250 align:middle line:84% So the dividing point of P bar is 00:20:47.250 --> 00:20:51.950 align:middle line:84% the union of the dividing points of the two partitions. 00:20:51.950 --> 00:20:59.810 align:middle line:84% But this means that P bar here-- so P bar here is a refinement. 00:20:59.810 --> 00:21:14.850 align:middle line:84% It's a refinement of subpartition of both P and P 00:21:14.850 --> 00:21:20.146 align:middle line:84% star, because it has all of the dividing points of either. 00:21:20.146 --> 00:21:22.450 align:middle line:90% But this means that-- 00:21:22.450 --> 00:21:26.630 align:middle line:90% remember that the lower sum-- 00:21:26.630 --> 00:21:28.870 align:middle line:90% so if you're looking at this-- 00:21:28.870 --> 00:21:35.230 align:middle line:84% the lower sum, if you take a subpartition, that goes up. 00:21:35.230 --> 00:21:41.390 align:middle line:84% So if you take a subpartition, the lower sum will go up. 00:21:41.390 --> 00:21:44.450 align:middle line:84% But the lower sum is always less than 00:21:44.450 --> 00:21:48.710 align:middle line:84% or equal to the upper sum for a given partition. 00:21:48.710 --> 00:21:53.910 align:middle line:84% But this here is now a refinement of P star. 00:21:53.910 --> 00:21:56.350 align:middle line:90% So the upper sum goes down. 00:21:56.350 --> 00:22:02.530 align:middle line:84% So this is less than or equal to U f of P star. 00:22:02.530 --> 00:22:05.190 align:middle line:84% And so there you have it, that this thing here, the lower 00:22:05.190 --> 00:22:09.350 align:middle line:84% sum for one partition, is always smaller than the upper sum 00:22:09.350 --> 00:22:12.070 align:middle line:90% for the other. 00:22:12.070 --> 00:22:15.930 align:middle line:84% And so this means that if you now take a partition-- 00:22:15.930 --> 00:22:26.210 align:middle line:90% 00:22:26.210 --> 00:22:37.330 align:middle line:84% so now it means that for a fixed partition P-- 00:22:37.330 --> 00:22:41.110 align:middle line:84% so if we have that this thing here is less than 00:22:41.110 --> 00:22:47.790 align:middle line:84% or equal to this thing about P as being fixed, 00:22:47.790 --> 00:22:49.690 align:middle line:90% but this is now for all other-- 00:22:49.690 --> 00:22:52.270 align:middle line:90% for all partitions P star. 00:22:52.270 --> 00:22:56.970 align:middle line:90% So this is for all P star. 00:22:56.970 --> 00:23:00.010 align:middle line:84% But if it holds for all, this is-- 00:23:00.010 --> 00:23:02.730 align:middle line:84% sorry-- is less than or equal to. 00:23:02.730 --> 00:23:06.710 align:middle line:84% If it holds for all, then it-- also, this inequality, 00:23:06.710 --> 00:23:08.170 align:middle line:90% this is a fixed number. 00:23:08.170 --> 00:23:13.210 align:middle line:84% This means that this here is a lower bound for this family. 00:23:13.210 --> 00:23:22.490 align:middle line:84% So this means that L f of P is less than or equal to the inf 00:23:22.490 --> 00:23:23.360 align:middle line:90% of those guys. 00:23:23.360 --> 00:23:30.270 align:middle line:90% 00:23:30.270 --> 00:23:32.870 align:middle line:84% It's less than or equal to the inf. 00:23:32.870 --> 00:23:35.350 align:middle line:90% But that here is just the-- 00:23:35.350 --> 00:23:41.430 align:middle line:84% this here is by definition the upper Riemann sum-- 00:23:41.430 --> 00:23:42.800 align:middle line:90% or, rather, Riemann integral. 00:23:42.800 --> 00:23:47.570 align:middle line:90% 00:23:47.570 --> 00:23:48.890 align:middle line:90% But this is for each. 00:23:48.890 --> 00:23:51.210 align:middle line:84% So now you forget about the P star. 00:23:51.210 --> 00:23:54.190 align:middle line:84% You have that this thing here is less than that number. 00:23:54.190 --> 00:23:56.230 align:middle line:84% So this means that this number here 00:23:56.230 --> 00:24:00.716 align:middle line:84% is an upper bound for this whole family. 00:24:00.716 --> 00:24:04.750 align:middle line:84% So this means that also the sup, the smallest upper bound, 00:24:04.750 --> 00:24:06.990 align:middle line:90% must be less than that. 00:24:06.990 --> 00:24:15.150 align:middle line:84% So this means that the sup of L of f, p 00:24:15.150 --> 00:24:18.450 align:middle line:84% must be less than or equal to this thing here. 00:24:18.450 --> 00:24:21.810 align:middle line:90% 00:24:21.810 --> 00:24:25.580 align:middle line:84% But the sup is by definition the lower integral. 00:24:25.580 --> 00:24:29.770 align:middle line:90% 00:24:29.770 --> 00:24:31.530 align:middle line:90% So you always have-- 00:24:31.530 --> 00:24:33.550 align:middle line:90% and this claim over here. 00:24:33.550 --> 00:24:38.290 align:middle line:90% 00:24:38.290 --> 00:24:44.890 align:middle line:90% So that's the first. 00:24:44.890 --> 00:24:53.450 align:middle line:84% So let's look at a function that is-- 00:24:53.450 --> 00:24:57.670 align:middle line:84% I added just a little bit, if you find it interesting. 00:24:57.670 --> 00:24:59.290 align:middle line:84% I find it interesting, but maybe you 00:24:59.290 --> 00:25:04.330 align:middle line:84% don't, but a little bit about Riemann, that he died 00:25:04.330 --> 00:25:08.290 align:middle line:90% rather young of tuberculosis. 00:25:08.290 --> 00:25:12.413 align:middle line:90% But he had an incredible impact. 00:25:12.413 --> 00:25:13.330 align:middle line:90% He had an incredible-- 00:25:13.330 --> 00:25:15.730 align:middle line:84% I mean, this is one of his contributions. 00:25:15.730 --> 00:25:20.350 align:middle line:84% His first contribution is he was a student 00:25:20.350 --> 00:25:26.750 align:middle line:84% of another really famous mathematician, perhaps 00:25:26.750 --> 00:25:28.290 align:middle line:90% the most famous of all time-- 00:25:28.290 --> 00:25:29.390 align:middle line:90% Gauss. 00:25:29.390 --> 00:25:35.550 align:middle line:84% And so at those times, when you wrote your thesis, 00:25:35.550 --> 00:25:38.250 align:middle line:84% then you had like six months to write your thesis. 00:25:38.250 --> 00:25:46.950 align:middle line:90% 00:25:46.950 --> 00:25:49.450 align:middle line:84% Formally, you couldn't choose your subject, 00:25:49.450 --> 00:25:51.510 align:middle line:90% but it was always-- 00:25:51.510 --> 00:25:54.130 align:middle line:84% anyway, you always got permission to choose it. 00:25:54.130 --> 00:25:57.350 align:middle line:84% You had to give three suggestions for something 00:25:57.350 --> 00:25:59.710 align:middle line:90% you would write your thesis on. 00:25:59.710 --> 00:26:01.350 align:middle line:90% But it was always-- 00:26:01.350 --> 00:26:05.390 align:middle line:84% it was a unspoken rule that your advisor always 00:26:05.390 --> 00:26:06.430 align:middle line:90% picked number one. 00:26:06.430 --> 00:26:07.570 align:middle line:90% And that's what you wanted. 00:26:07.570 --> 00:26:09.990 align:middle line:84% And then you could add some other crazy stuff 00:26:09.990 --> 00:26:12.670 align:middle line:90% as two other suggestions. 00:26:12.670 --> 00:26:15.830 align:middle line:84% So Riemann had three suggestions. 00:26:15.830 --> 00:26:21.330 align:middle line:84% And the third one, which he had absolutely no idea about, 00:26:21.330 --> 00:26:27.010 align:middle line:84% was to make sense of curvature in higher dimension. 00:26:27.010 --> 00:26:30.110 align:middle line:90% And so he just put it in there. 00:26:30.110 --> 00:26:33.130 align:middle line:84% He knew it wasn't going to be selected. 00:26:33.130 --> 00:26:35.330 align:middle line:84% But actually, at the time, this was 00:26:35.330 --> 00:26:37.250 align:middle line:90% what Gauss was interested in. 00:26:37.250 --> 00:26:39.430 align:middle line:84% And so Gauss said, to hell with the rules. 00:26:39.430 --> 00:26:41.010 align:middle line:90% I'll pick number three. 00:26:41.010 --> 00:26:44.150 align:middle line:84% And so he had six months, and nobody had figured this out. 00:26:44.150 --> 00:26:47.210 align:middle line:84% So Gauss is considered by many the greatest 00:26:47.210 --> 00:26:50.370 align:middle line:90% mathematician of all time. 00:26:50.370 --> 00:26:53.010 align:middle line:84% But after six months, actually, so Riemann 00:26:53.010 --> 00:26:55.930 align:middle line:84% came up with something called the curvature tensor, 00:26:55.930 --> 00:26:59.290 align:middle line:84% which is this objects in all dimensions. 00:26:59.290 --> 00:27:02.090 align:middle line:84% And that generalized Gauss's work. 00:27:02.090 --> 00:27:06.090 align:middle line:84% And this lay the foundation for general relativity 00:27:06.090 --> 00:27:14.770 align:middle line:84% that came many years later, 70 years or so later. 00:27:14.770 --> 00:27:16.970 align:middle line:90% So that was the Riemann's first. 00:27:16.970 --> 00:27:20.230 align:middle line:84% This is one of Riemann's other contributions. 00:27:20.230 --> 00:27:23.990 align:middle line:84% And there are other things that was only published a long time 00:27:23.990 --> 00:27:29.110 align:middle line:84% after his death that have had a tremendous contribution 00:27:29.110 --> 00:27:31.070 align:middle line:90% in other areas. 00:27:31.070 --> 00:27:33.070 align:middle line:90% Anyway, so this is-- 00:27:33.070 --> 00:27:36.450 align:middle line:84% so this was Riemann's way of defining the integrals. 00:27:36.450 --> 00:27:44.830 align:middle line:90% 00:27:44.830 --> 00:27:48.312 align:middle line:90% Let me look at a function. 00:27:48.312 --> 00:27:49.770 align:middle line:84% So we talk about that a little bit. 00:27:49.770 --> 00:27:52.770 align:middle line:84% But we didn't actually define it. 00:27:52.770 --> 00:27:56.870 align:middle line:84% So here's an example of a function 00:27:56.870 --> 00:27:59.870 align:middle line:90% that is not Riemann integrable. 00:27:59.870 --> 00:28:04.750 align:middle line:84% So it's a function defined on the interval from 0 to 1, 00:28:04.750 --> 00:28:07.790 align:middle line:90% so it's bounded. 00:28:07.790 --> 00:28:18.970 align:middle line:84% And so the function is f of x is equal to 1 00:28:18.970 --> 00:28:21.600 align:middle line:90% if x here is rational. 00:28:21.600 --> 00:28:25.650 align:middle line:90% 00:28:25.650 --> 00:28:28.190 align:middle line:90% And it's 0 otherwise. 00:28:28.190 --> 00:28:35.258 align:middle line:90% 00:28:35.258 --> 00:28:39.510 align:middle line:84% And so of course, this function here is always between 0 and 1. 00:28:39.510 --> 00:28:42.610 align:middle line:90% So it's definitely bounded. 00:28:42.610 --> 00:28:45.190 align:middle line:84% But if you take now any interval-- 00:28:45.190 --> 00:28:50.530 align:middle line:90% 00:28:50.530 --> 00:29:03.950 align:middle line:84% so this here, so I claim here f is not Riemann integrable. 00:29:03.950 --> 00:29:13.290 align:middle line:90% 00:29:13.290 --> 00:29:15.620 align:middle line:84% And so the thing is that if you take-- 00:29:15.620 --> 00:29:19.070 align:middle line:90% 00:29:19.070 --> 00:29:25.070 align:middle line:84% if you take a interval like this and you 00:29:25.070 --> 00:29:34.610 align:middle line:84% take a subpartition like that, and now you have that-- 00:29:34.610 --> 00:29:39.070 align:middle line:84% and now you're looking at one of these mi. 00:29:39.070 --> 00:29:41.770 align:middle line:90% So here, you have xi minus 1. 00:29:41.770 --> 00:29:44.910 align:middle line:90% And here you have xi. 00:29:44.910 --> 00:29:48.670 align:middle line:90% And so you're looking at mi. 00:29:48.670 --> 00:29:51.830 align:middle line:84% So this is some little interval like this. 00:29:51.830 --> 00:29:55.070 align:middle line:84% But on any kind of small interval, 00:29:55.070 --> 00:29:57.010 align:middle line:84% if it's really a proper integral, 00:29:57.010 --> 00:29:58.510 align:middle line:84% if it's not just a point but it's 00:29:58.510 --> 00:30:01.950 align:middle line:84% a proper integral-- interval, then there's 00:30:01.950 --> 00:30:04.790 align:middle line:84% always points that are rational and always points 00:30:04.790 --> 00:30:06.150 align:middle line:90% that are not rational. 00:30:06.150 --> 00:30:10.950 align:middle line:84% So this means that this here, this here, the sup here, 00:30:10.950 --> 00:30:17.930 align:middle line:84% would have to be 1, but the inf here would be 0. 00:30:17.930 --> 00:30:20.080 align:middle line:90% And so this means that for the-- 00:30:20.080 --> 00:30:22.030 align:middle line:90% so this is just any partition. 00:30:22.030 --> 00:30:24.250 align:middle line:90% And so this means that for-- 00:30:24.250 --> 00:30:29.610 align:middle line:84% if you're looking at the upper sum here, 00:30:29.610 --> 00:30:34.290 align:middle line:84% but it's now the sum of these capital Mi delta xi. 00:30:34.290 --> 00:30:35.730 align:middle line:90% This here is 1. 00:30:35.730 --> 00:30:39.450 align:middle line:84% So it's just the sum of the length of the small interval. 00:30:39.450 --> 00:30:43.730 align:middle line:84% But that's just one because the whole interval is from 0 to 1. 00:30:43.730 --> 00:30:48.650 align:middle line:84% And likewise, for the lower interval, lower sum, 00:30:48.650 --> 00:30:55.450 align:middle line:84% this is the sum here of this small mi delta xi. 00:30:55.450 --> 00:30:56.970 align:middle line:90% These here are 0. 00:30:56.970 --> 00:30:58.790 align:middle line:90% So this whole thing here is 0. 00:30:58.790 --> 00:31:01.730 align:middle line:90% 00:31:01.730 --> 00:31:08.370 align:middle line:84% So you see that if you-- this is for all partitions. 00:31:08.370 --> 00:31:11.170 align:middle line:84% So this means that if you take the infimum of this, 00:31:11.170 --> 00:31:12.290 align:middle line:90% you still get 1. 00:31:12.290 --> 00:31:15.510 align:middle line:84% And if you take the sup of these still get 0. 00:31:15.510 --> 00:31:18.530 align:middle line:84% So those two numbers are not the same. 00:31:18.530 --> 00:31:22.870 align:middle line:84% The lower and the upper Riemann integrals are not the same. 00:31:22.870 --> 00:31:25.170 align:middle line:84% So the function is not Riemann integrable. 00:31:25.170 --> 00:31:32.430 align:middle line:90% 00:31:32.430 --> 00:31:34.920 align:middle line:90% So now the next thing is-- 00:31:34.920 --> 00:31:38.390 align:middle line:90% 00:31:38.390 --> 00:31:42.630 align:middle line:84% the next thing is-- so we want to know which functions 00:31:42.630 --> 00:31:45.710 align:middle line:90% are Riemann integrable. 00:31:45.710 --> 00:31:49.950 align:middle line:84% And so here's a useful little criteria. 00:31:49.950 --> 00:31:56.220 align:middle line:84% So lemma-- so you have a function again that is bounded. 00:31:56.220 --> 00:32:00.990 align:middle line:90% 00:32:00.990 --> 00:32:08.170 align:middle line:90% Then f is Riemann integrable. 00:32:08.170 --> 00:32:11.210 align:middle line:84% And again, we write it like this. 00:32:11.210 --> 00:32:20.770 align:middle line:84% So it's Riemann integrable if for all epsilon greater than 0, 00:32:20.770 --> 00:32:40.850 align:middle line:84% there exists a partition P so that the upper integral-- 00:32:40.850 --> 00:32:46.390 align:middle line:84% upper sum-- sorry-- so that the upper sum minus the lower sum-- 00:32:46.390 --> 00:32:51.050 align:middle line:84% remember that this is always bigger than the lower sum 00:32:51.050 --> 00:32:55.410 align:middle line:84% if this here is less than epsilon. 00:32:55.410 --> 00:32:59.130 align:middle line:84% So if we want to check that a function is Riemann integrable, 00:32:59.130 --> 00:33:01.890 align:middle line:84% then we just need to check that for all epsilon, 00:33:01.890 --> 00:33:09.390 align:middle line:84% we can find a partition with this property. 00:33:09.390 --> 00:33:13.830 align:middle line:84% And so this is a statement that is if and only if-- so I should 00:33:13.830 --> 00:33:15.410 align:middle line:90% say if and only if-- 00:33:15.410 --> 00:33:18.430 align:middle line:90% 00:33:18.430 --> 00:33:21.650 align:middle line:90% if and only if. 00:33:21.650 --> 00:33:35.310 align:middle line:90% 00:33:35.310 --> 00:33:37.030 align:middle line:90% So just try to prove that. 00:33:37.030 --> 00:33:40.790 align:middle line:84% And then we're going to use this later on 00:33:40.790 --> 00:33:45.770 align:middle line:84% to prove that a lot of functions are Riemann integrable. 00:33:45.770 --> 00:33:50.190 align:middle line:90% 00:33:50.190 --> 00:33:59.250 align:middle line:90% So suppose that f-- 00:33:59.250 --> 00:34:03.530 align:middle line:84% so if f here is Riemann integrable-- 00:34:03.530 --> 00:34:11.889 align:middle line:90% 00:34:11.889 --> 00:34:14.290 align:middle line:84% so if it is Riemann integrable, then we 00:34:14.290 --> 00:34:20.138 align:middle line:84% know that this lower Riemann integral 00:34:20.138 --> 00:34:23.099 align:middle line:84% is equal to the upper Riemann integral. 00:34:23.099 --> 00:34:29.170 align:middle line:90% 00:34:29.170 --> 00:34:32.210 align:middle line:84% This here was the sup of all-- this here 00:34:32.210 --> 00:34:39.050 align:middle line:84% was the sup of all lower sums for all-- over all partitions. 00:34:39.050 --> 00:34:48.170 align:middle line:84% And this here was the inf of all upper sum, 00:34:48.170 --> 00:34:50.989 align:middle line:90% like that, over all partitions. 00:34:50.989 --> 00:34:57.930 align:middle line:90% 00:34:57.930 --> 00:35:08.550 align:middle line:84% So therefore, given epsilon greater than 0, 00:35:08.550 --> 00:35:20.350 align:middle line:90% there exists a partition P1. 00:35:20.350 --> 00:35:20.850 align:middle line:90% Why? 00:35:20.850 --> 00:35:23.050 align:middle line:84% Because this here has a sup over all partitions. 00:35:23.050 --> 00:35:31.870 align:middle line:84% So it means that there exists a partition so that this lower 00:35:31.870 --> 00:35:46.010 align:middle line:84% sum here is bigger or equal to this here minus epsilon over 2. 00:35:46.010 --> 00:35:49.010 align:middle line:84% It's just because it's a sup of all these numbers. 00:35:49.010 --> 00:35:54.220 align:middle line:84% So there must exist one that comes this close to the sup. 00:35:54.220 --> 00:36:11.330 align:middle line:90% 00:36:11.330 --> 00:36:24.410 align:middle line:84% Likewise, there exists a partition P2-- 00:36:24.410 --> 00:36:26.930 align:middle line:90% sorry-- this here should be P1. 00:36:26.930 --> 00:36:29.370 align:middle line:90% This is the sup over-- sorry. 00:36:29.370 --> 00:36:33.990 align:middle line:84% So there exists a partition P2 so that the-- 00:36:33.990 --> 00:36:39.290 align:middle line:84% so that for that, this is one where this value here 00:36:39.290 --> 00:36:40.270 align:middle line:90% is really small. 00:36:40.270 --> 00:36:43.670 align:middle line:84% So this is not too much larger than this. 00:36:43.670 --> 00:36:54.370 align:middle line:84% So this means that this for P2 is smaller or equal to this 00:36:54.370 --> 00:37:02.870 align:middle line:84% here, upper Riemann integral plus epsilon over 2. 00:37:02.870 --> 00:37:09.470 align:middle line:90% 00:37:09.470 --> 00:37:32.750 align:middle line:84% So now let P be the partition with all the dividing points 00:37:32.750 --> 00:37:40.130 align:middle line:90% of both P1 and P2. 00:37:40.130 --> 00:37:43.470 align:middle line:84% So it has all of the dividing points from P1 00:37:43.470 --> 00:37:46.936 align:middle line:84% together with all of the dividing points from P2. 00:37:46.936 --> 00:37:50.910 align:middle line:84% But now for this partition, we have that-- 00:37:50.910 --> 00:37:55.310 align:middle line:84% well, for-- this is a refinement of P1. 00:37:55.310 --> 00:37:58.610 align:middle line:84% So if it's refinement, the value here goes up. 00:37:58.610 --> 00:38:03.330 align:middle line:90% 00:38:03.330 --> 00:38:06.370 align:middle line:90% And this thing here is-- 00:38:06.370 --> 00:38:11.730 align:middle line:84% so this is less than the upper for the same thing. 00:38:11.730 --> 00:38:15.610 align:middle line:84% And since it's a refinement also of P2, 00:38:15.610 --> 00:38:20.150 align:middle line:84% we have that it is less than the upper sum for P2. 00:38:20.150 --> 00:38:23.530 align:middle line:90% 00:38:23.530 --> 00:38:28.210 align:middle line:84% But now you see that this thing here-- 00:38:28.210 --> 00:38:32.750 align:middle line:84% so L-- so the difference between these two-- 00:38:32.750 --> 00:38:35.330 align:middle line:84% the difference between these two numbers-- 00:38:35.330 --> 00:38:37.470 align:middle line:90% so this is the difference here. 00:38:37.470 --> 00:38:45.610 align:middle line:90% So U f P minus U f P-- sorry-- 00:38:45.610 --> 00:38:51.870 align:middle line:84% L f P, this difference here, this is bigger than that. 00:38:51.870 --> 00:38:54.050 align:middle line:84% So if you take this and subtract this, 00:38:54.050 --> 00:38:56.210 align:middle line:84% you're getting something non-negative. 00:38:56.210 --> 00:38:59.190 align:middle line:84% And this, the spread between these two, 00:38:59.190 --> 00:39:02.110 align:middle line:84% is smaller than the spread between these. 00:39:02.110 --> 00:39:10.520 align:middle line:84% So this is less than U of f P2 minus L of f P1. 00:39:10.520 --> 00:39:13.030 align:middle line:90% 00:39:13.030 --> 00:39:16.750 align:middle line:90% But this thing here is-- 00:39:16.750 --> 00:39:25.590 align:middle line:84% this thing here is less than U f P2 minus this upper integral 00:39:25.590 --> 00:39:26.090 align:middle line:90% here. 00:39:26.090 --> 00:39:30.190 align:middle line:90% 00:39:30.190 --> 00:39:31.890 align:middle line:84% Assuming that it's Riemann integral, 00:39:31.890 --> 00:39:34.230 align:middle line:84% so the upper and lower are the same. 00:39:34.230 --> 00:39:47.870 align:middle line:84% So this thing here we can write it as this plus minus L f P1 00:39:47.870 --> 00:39:56.730 align:middle line:84% minus the integral here, like that. 00:39:56.730 --> 00:40:02.920 align:middle line:84% I'm just subtracting and then adding back the same thing. 00:40:02.920 --> 00:40:06.320 align:middle line:84% And this one here, we already found that this here 00:40:06.320 --> 00:40:10.780 align:middle line:90% was bounded by epsilon over 2. 00:40:10.780 --> 00:40:12.880 align:middle line:84% This here is-- remember, we're assuming 00:40:12.880 --> 00:40:15.020 align:middle line:90% that it's Riemann integrable. 00:40:15.020 --> 00:40:20.640 align:middle line:84% So this here is the same thing as when you have the upper one. 00:40:20.640 --> 00:40:24.000 align:middle line:90% And this thing here is-- 00:40:24.000 --> 00:40:29.180 align:middle line:84% this here is also bounded by-- this is actually right. 00:40:29.180 --> 00:40:33.700 align:middle line:90% This is negative, but it's-- 00:40:33.700 --> 00:40:40.680 align:middle line:84% so this thing here is bigger or equal to-- 00:40:40.680 --> 00:40:42.070 align:middle line:90% maybe I should write it as-- 00:40:42.070 --> 00:40:45.600 align:middle line:90% 00:40:45.600 --> 00:40:48.280 align:middle line:84% well, let me write it like this-- plus. 00:40:48.280 --> 00:40:51.680 align:middle line:84% And then I can write it as the integral from a 00:40:51.680 --> 00:40:59.820 align:middle line:84% to b of f of x dx minus L f of P1. 00:40:59.820 --> 00:41:04.420 align:middle line:84% But this thing here was bounded by epsilon over 2 also. 00:41:04.420 --> 00:41:08.020 align:middle line:84% So epsilon over 2 plus epsilon over 2. 00:41:08.020 --> 00:41:10.900 align:middle line:90% So this is equal to epsilon. 00:41:10.900 --> 00:41:15.940 align:middle line:90% 00:41:15.940 --> 00:41:21.460 align:middle line:84% So again, if you want to test that a function is Riemann 00:41:21.460 --> 00:41:24.560 align:middle line:84% integral, you just need to find-- for a given, 00:41:24.560 --> 00:41:26.620 align:middle line:84% for any given epsilon, you need to be 00:41:26.620 --> 00:41:32.740 align:middle line:84% able to find a partition so that the upper sum for that partition 00:41:32.740 --> 00:41:37.540 align:middle line:84% is minus the lower sum for the same partition 00:41:37.540 --> 00:41:38.680 align:middle line:90% is less than epsilon. 00:41:38.680 --> 00:41:42.495 align:middle line:90% 00:41:42.495 --> 00:41:47.740 align:middle line:84% And so now this leads us to the next theorem. 00:41:47.740 --> 00:41:52.920 align:middle line:84% So the next theorem is that any continuous function 00:41:52.920 --> 00:41:54.580 align:middle line:90% is Riemann integrable. 00:41:54.580 --> 00:42:07.280 align:middle line:90% 00:42:07.280 --> 00:42:09.240 align:middle line:90% So any Riemann sum-- 00:42:09.240 --> 00:42:10.840 align:middle line:90% so any continuous function-- 00:42:10.840 --> 00:42:20.480 align:middle line:90% 00:42:20.480 --> 00:42:26.600 align:middle line:84% any continuous function on an integral like this-- this is-- 00:42:26.600 --> 00:42:30.235 align:middle line:90% a and b are real numbers-- 00:42:30.235 --> 00:42:33.180 align:middle line:90% it is Riemann integrable. 00:42:33.180 --> 00:42:41.800 align:middle line:90% 00:42:41.800 --> 00:42:45.480 align:middle line:84% So this is the next theorem that we want to prove. 00:42:45.480 --> 00:42:48.720 align:middle line:84% But in order to prove that, we need a tool. 00:42:48.720 --> 00:42:52.800 align:middle line:84% And so this is what's called uniformly continuous. 00:42:52.800 --> 00:42:56.620 align:middle line:90% 00:42:56.620 --> 00:42:59.260 align:middle line:84% So just to show this, we need the notion 00:42:59.260 --> 00:43:03.420 align:middle line:90% of being uniformly continuous. 00:43:03.420 --> 00:43:06.500 align:middle line:90% So what is that? 00:43:06.500 --> 00:43:30.600 align:middle line:84% So in the proof, we need the notion of uniformly continuous. 00:43:30.600 --> 00:43:39.820 align:middle line:90% 00:43:39.820 --> 00:43:42.260 align:middle line:90% So that's the following. 00:43:42.260 --> 00:43:47.060 align:middle line:84% And you have a function defined on some interval, say. 00:43:47.060 --> 00:43:51.720 align:middle line:84% It could be all of R, for that matter. 00:43:51.720 --> 00:43:54.440 align:middle line:84% So you have a function like this. 00:43:54.440 --> 00:43:58.200 align:middle line:90% And f is said to be-- 00:43:58.200 --> 00:44:21.480 align:middle line:84% is said to be uniformly continuous if for all epsilon 00:44:21.480 --> 00:44:31.000 align:middle line:84% greater than 0, there exists delta greater than 0 00:44:31.000 --> 00:44:39.200 align:middle line:84% such that as long as x and y are less than delta, then 00:44:39.200 --> 00:44:48.344 align:middle line:84% their images is less than epsilon. 00:44:48.344 --> 00:44:52.120 align:middle line:84% So what is the difference between this and continuity? 00:44:52.120 --> 00:44:54.940 align:middle line:84% For continuity, you're always-- you're 00:44:54.940 --> 00:44:57.440 align:middle line:84% looking at-- a function's continuous 00:44:57.440 --> 00:45:00.160 align:middle line:84% if it's continuous at all points. 00:45:00.160 --> 00:45:02.100 align:middle line:84% And so it's continuous at a point. 00:45:02.100 --> 00:45:05.220 align:middle line:90% 00:45:05.220 --> 00:45:06.620 align:middle line:84% To be continuous at a point, it's 00:45:06.620 --> 00:45:08.940 align:middle line:84% like you're fixing one of these two. 00:45:08.940 --> 00:45:11.300 align:middle line:90% So it's like x is fixed. 00:45:11.300 --> 00:45:15.220 align:middle line:84% And then you're looking at if y is sufficiently close, then 00:45:15.220 --> 00:45:16.620 align:middle line:90% the value-- 00:45:16.620 --> 00:45:20.020 align:middle line:84% the difference between the value at x and y 00:45:20.020 --> 00:45:21.660 align:middle line:90% is smaller than epsilon. 00:45:21.660 --> 00:45:26.880 align:middle line:84% But here, it's like that the same delta works for all x. 00:45:26.880 --> 00:45:30.380 align:middle line:84% You don't have to fix one of the two points. 00:45:30.380 --> 00:45:31.800 align:middle line:90% The same delta work. 00:45:31.800 --> 00:45:38.100 align:middle line:84% So let's see where this is not the case. 00:45:38.100 --> 00:45:41.340 align:middle line:90% So let me look at two examples. 00:45:41.340 --> 00:45:43.360 align:middle line:84% So just to understand the difference-- 00:45:43.360 --> 00:45:46.200 align:middle line:84% so to understand the difference-- 00:45:46.200 --> 00:45:51.200 align:middle line:90% 00:45:51.200 --> 00:46:05.740 align:middle line:84% the difference between continuity and uniform 00:46:05.740 --> 00:46:07.400 align:middle line:90% continuity. 00:46:07.400 --> 00:46:14.380 align:middle line:84% Uniform, again, refer to that x is not fixed, or x0. 00:46:14.380 --> 00:46:16.680 align:middle line:90% Usually, we call it x0 and x. 00:46:16.680 --> 00:46:20.500 align:middle line:84% So one of the two points was fixed. 00:46:20.500 --> 00:46:23.340 align:middle line:84% And uniform means that you don't need to fix. 00:46:23.340 --> 00:46:28.080 align:middle line:84% It's like the same delta works for all x0's. 00:46:28.080 --> 00:46:33.880 align:middle line:90% 00:46:33.880 --> 00:46:41.160 align:middle line:84% And so here is an example that suppose you take the function f 00:46:41.160 --> 00:46:42.820 align:middle line:90% of x equal to x squared. 00:46:42.820 --> 00:46:48.780 align:middle line:84% This is now on R. Well, we already 00:46:48.780 --> 00:46:51.420 align:middle line:84% know that this function is continuous. 00:46:51.420 --> 00:47:04.500 align:middle line:84% And so if it was uniformly so, if f was uniformly continuous, 00:47:04.500 --> 00:47:12.820 align:middle line:84% then for a given epsilon greater than 0, 00:47:12.820 --> 00:47:18.460 align:middle line:84% there would exist a delta so that the value at x 00:47:18.460 --> 00:47:26.020 align:middle line:84% and the value where you, say, added delta, 00:47:26.020 --> 00:47:29.460 align:middle line:84% this difference here would have to be 00:47:29.460 --> 00:47:33.040 align:middle line:84% less than if you want-- if you don't like it, 00:47:33.040 --> 00:47:36.860 align:middle line:84% you could say delta over 2 if you want it to be strictly 00:47:36.860 --> 00:47:38.188 align:middle line:90% the difference between. 00:47:38.188 --> 00:47:39.480 align:middle line:90% But that doesn't really matter. 00:47:39.480 --> 00:47:42.100 align:middle line:90% I'll just say like that. 00:47:42.100 --> 00:47:44.920 align:middle line:84% This here would have to be the case. 00:47:44.920 --> 00:47:47.940 align:middle line:84% And this here would have to be the case for all x. 00:47:47.940 --> 00:47:52.160 align:middle line:90% 00:47:52.160 --> 00:47:54.080 align:middle line:90% But what is this here? 00:47:54.080 --> 00:47:58.620 align:middle line:84% This here is x plus delta squared. 00:47:58.620 --> 00:48:01.560 align:middle line:84% That's the value of the function in this. 00:48:01.560 --> 00:48:04.300 align:middle line:84% And this is the value of the function in x. 00:48:04.300 --> 00:48:07.380 align:middle line:90% 00:48:07.380 --> 00:48:09.760 align:middle line:84% But this thing here, it's a squaring out. 00:48:09.760 --> 00:48:17.880 align:middle line:84% It gives you x squared plus delta squared plus 2x delta. 00:48:17.880 --> 00:48:22.090 align:middle line:84% And this thing here would have to be less than epsilon. 00:48:22.090 --> 00:48:25.704 align:middle line:90% 00:48:25.704 --> 00:48:29.840 align:middle line:90% Delta is positive. 00:48:29.840 --> 00:48:30.400 align:middle line:90% Sorry. 00:48:30.400 --> 00:48:33.680 align:middle line:84% This here, minus x squared, would have 00:48:33.680 --> 00:48:35.160 align:middle line:90% to be less than delta epsilon. 00:48:35.160 --> 00:48:38.760 align:middle line:90% 00:48:38.760 --> 00:48:43.500 align:middle line:90% And so you get rid of these two. 00:48:43.500 --> 00:48:45.660 align:middle line:90% Delta is positive. 00:48:45.660 --> 00:48:50.440 align:middle line:84% So if you have this, you would definitely also have that 00:48:50.440 --> 00:48:53.580 align:middle line:84% so this would imply that-- or delta-- 00:48:53.580 --> 00:48:56.460 align:middle line:84% this here, delta square, I guess, is positive regardless. 00:48:56.460 --> 00:49:01.020 align:middle line:84% But you would have that 2x delta here, 00:49:01.020 --> 00:49:03.710 align:middle line:84% would have to be less than epsilon. 00:49:03.710 --> 00:49:06.900 align:middle line:90% 00:49:06.900 --> 00:49:09.120 align:middle line:84% Now I'm using the delta is positive. 00:49:09.120 --> 00:49:12.120 align:middle line:84% So this here would have to be the case for all x. 00:49:12.120 --> 00:49:14.780 align:middle line:90% 00:49:14.780 --> 00:49:19.220 align:middle line:84% But if you just let x be delta is now supposed to be-- 00:49:19.220 --> 00:49:21.700 align:middle line:84% you're supposed to find some delta so that 00:49:21.700 --> 00:49:23.120 align:middle line:90% this here works for all x. 00:49:23.120 --> 00:49:25.500 align:middle line:84% So this means that delta is fixed. 00:49:25.500 --> 00:49:28.580 align:middle line:84% But if now delta x becomes incredibly large, then 00:49:28.580 --> 00:49:31.920 align:middle line:84% of course, you don't have this inequality anymore. 00:49:31.920 --> 00:49:36.340 align:middle line:84% So this function is not uniform continuous. 00:49:36.340 --> 00:49:37.100 align:middle line:90% Yeah? 00:49:37.100 --> 00:49:38.300 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:49:38.300 --> 00:49:41.160 align:middle line:84% TOBIAS COLDING: Because it only helps us. 00:49:41.160 --> 00:49:47.260 align:middle line:84% So if this here is less than that, this is positive. 00:49:47.260 --> 00:49:52.612 align:middle line:84% This means that this here surely is also less than epsilon. 00:49:52.612 --> 00:49:53.950 align:middle line:90% So that's one example. 00:49:53.950 --> 00:49:58.440 align:middle line:90% 00:49:58.440 --> 00:50:02.440 align:middle line:90% The other example is-- 00:50:02.440 --> 00:50:08.160 align:middle line:84% and actually, let me draw a picture to just illustrate it. 00:50:08.160 --> 00:50:10.540 align:middle line:90% So if you take. 00:50:10.540 --> 00:50:13.840 align:middle line:84% So again the function roughly looks like this. 00:50:13.840 --> 00:50:16.830 align:middle line:84% This is f of x equal to x squared. 00:50:16.830 --> 00:50:20.920 align:middle line:90% 00:50:20.920 --> 00:50:26.360 align:middle line:84% This meaning of that it's uniform continuous means that 00:50:26.360 --> 00:50:32.640 align:middle line:84% if you take some epsilon for any given epsilon, 00:50:32.640 --> 00:50:38.240 align:middle line:84% so if you're looking at a interval over on the y-axis 00:50:38.240 --> 00:50:42.580 align:middle line:90% here, so this is between-- 00:50:42.580 --> 00:50:46.340 align:middle line:84% this here is like a epsilon interval. 00:50:46.340 --> 00:50:48.500 align:middle line:90% And you're looking at-- 00:50:48.500 --> 00:50:51.740 align:middle line:84% so if you're looking at some point here, 00:50:51.740 --> 00:50:54.260 align:middle line:90% this here is this point here-- 00:50:54.260 --> 00:50:57.220 align:middle line:90% is taken to this here. 00:50:57.220 --> 00:51:01.240 align:middle line:84% And now you want to look at some small interval. 00:51:01.240 --> 00:51:03.560 align:middle line:84% So that you end up in that interval. 00:51:03.560 --> 00:51:06.940 align:middle line:84% And the thing is that as you're moving further and further 00:51:06.940 --> 00:51:09.780 align:middle line:84% out this-- the length of that interval 00:51:09.780 --> 00:51:12.200 align:middle line:84% would have to be smaller and smaller. 00:51:12.200 --> 00:51:14.320 align:middle line:84% But you can't use the same uniform size. 00:51:14.320 --> 00:51:19.300 align:middle line:84% So that's why it's not uniformly continuous. 00:51:19.300 --> 00:51:23.740 align:middle line:84% The other example I want to look at 00:51:23.740 --> 00:51:26.300 align:middle line:90% is where you take a function-- 00:51:26.300 --> 00:51:29.730 align:middle line:90% 00:51:29.730 --> 00:51:34.700 align:middle line:84% where you're taking f of x equal to 1 over x, and x 00:51:34.700 --> 00:51:40.800 align:middle line:84% here is now in the interval, from 0 to 1. 00:51:40.800 --> 00:51:45.120 align:middle line:90% 1 is included, but 0 is not. 00:51:45.120 --> 00:51:52.120 align:middle line:84% And so there, the picture is that this here is the function. 00:51:52.120 --> 00:51:57.680 align:middle line:84% And so you see that as you're getting closer, here is 1. 00:51:57.680 --> 00:52:00.800 align:middle line:84% And as you're looking at things closer 00:52:00.800 --> 00:52:06.460 align:middle line:84% to 0, then even a really small interval, the functions, 00:52:06.460 --> 00:52:10.160 align:middle line:84% the value of the function would deviate a lot 00:52:10.160 --> 00:52:12.700 align:middle line:90% on even a very small interval. 00:52:12.700 --> 00:52:17.120 align:middle line:84% So you couldn't use the same interval size 00:52:17.120 --> 00:52:21.040 align:middle line:84% to say that the value of the function 00:52:21.040 --> 00:52:22.700 align:middle line:90% is smaller than some epsilon. 00:52:22.700 --> 00:52:25.480 align:middle line:90% So just make that precise. 00:52:25.480 --> 00:52:34.480 align:middle line:84% If you take a sequence xn equal to 1 over n and yn equal to 1 00:52:34.480 --> 00:52:37.820 align:middle line:90% over 2n, then-- 00:52:37.820 --> 00:52:39.010 align:middle line:90% and you're looking at-- 00:52:39.010 --> 00:52:43.900 align:middle line:90% 00:52:43.900 --> 00:52:46.340 align:middle line:84% you're looking at the difference, 00:52:46.340 --> 00:52:55.012 align:middle line:84% so if you're looking at xn minus 1, here, you have-- 00:52:55.012 --> 00:52:56.720 align:middle line:84% we don't even need to calculate anything. 00:52:56.720 --> 00:52:58.860 align:middle line:84% For the first thing here, you have xn. 00:52:58.860 --> 00:53:01.220 align:middle line:90% Here, you have yn. 00:53:01.220 --> 00:53:03.820 align:middle line:84% So this means that the difference between these two 00:53:03.820 --> 00:53:06.980 align:middle line:84% is certainly bounded by this here. 00:53:06.980 --> 00:53:09.300 align:middle line:90% So of course, you always have-- 00:53:09.300 --> 00:53:11.040 align:middle line:84% you have actually something much better. 00:53:11.040 --> 00:53:13.440 align:middle line:84% But this is actually smaller than 1 over n. 00:53:13.440 --> 00:53:16.260 align:middle line:90% 00:53:16.260 --> 00:53:21.860 align:middle line:84% And if you're now looking at the value of the function, 00:53:21.860 --> 00:53:25.540 align:middle line:90% this thing here is 1 over xn. 00:53:25.540 --> 00:53:27.780 align:middle line:90% So that's n. 00:53:27.780 --> 00:53:34.500 align:middle line:90% And f of yn is 1 over 2n. 00:53:34.500 --> 00:53:36.280 align:middle line:90% So this is 2n. 00:53:36.280 --> 00:53:44.680 align:middle line:84% So this means that f of xn minus f of yn here, 00:53:44.680 --> 00:53:50.905 align:middle line:84% this here is equal to this difference here, is equal to 2n. 00:53:50.905 --> 00:53:54.760 align:middle line:90% 00:53:54.760 --> 00:53:58.088 align:middle line:84% And so you see that you couldn't-- 00:53:58.088 --> 00:54:02.440 align:middle line:84% it's not like-- it's not-- for any given epsilon, 00:54:02.440 --> 00:54:06.560 align:middle line:84% it doesn't work that if these two points are, 00:54:06.560 --> 00:54:09.000 align:middle line:84% doesn't exist a delta so that if these two 00:54:09.000 --> 00:54:15.420 align:middle line:84% points are delta closed, then the images is at most epsilon. 00:54:15.420 --> 00:54:15.920 align:middle line:90% Yeah? 00:54:15.920 --> 00:54:16.800 align:middle line:84% AUDIENCE: I don't know if this is too broad 00:54:16.800 --> 00:54:19.467 align:middle line:84% of a generalization, but is that essentially just the same thing 00:54:19.467 --> 00:54:21.558 align:middle line:84% as saying that the slope is bounded. 00:54:21.558 --> 00:54:23.100 align:middle line:84% TOBIAS COLDING: Yeah, it's like that. 00:54:23.100 --> 00:54:24.740 align:middle line:90% I mean, it's sort of like that. 00:54:24.740 --> 00:54:25.640 align:middle line:90% I mean, it's not-- 00:54:25.640 --> 00:54:31.240 align:middle line:84% I mean, uniform continuity doesn't require-- 00:54:31.240 --> 00:54:35.566 align:middle line:84% I mean, sort of like that but not exactly, because uniformly-- 00:54:35.566 --> 00:54:39.760 align:middle line:84% slope is something about being differentiable, 00:54:39.760 --> 00:54:44.020 align:middle line:84% and you can have a uniformly continuous function 00:54:44.020 --> 00:54:45.480 align:middle line:90% that is not differentiable. 00:54:45.480 --> 00:54:48.180 align:middle line:90% 00:54:48.180 --> 00:54:50.080 align:middle line:90% But it's definitely related. 00:54:50.080 --> 00:54:50.580 align:middle line:90% Yeah. 00:54:50.580 --> 00:54:54.680 align:middle line:90% 00:54:54.680 --> 00:54:59.860 align:middle line:84% So anyway, so uniform again means that for a given delta-- 00:54:59.860 --> 00:55:05.300 align:middle line:84% for a given epsilon, you can use the same delta for-- 00:55:05.300 --> 00:55:06.920 align:middle line:90% at all points. 00:55:06.920 --> 00:55:09.060 align:middle line:84% Otherwise, for continuity, you are always 00:55:09.060 --> 00:55:10.280 align:middle line:90% fixing one of the points. 00:55:10.280 --> 00:55:13.580 align:middle line:84% And then you were looking at other points 00:55:13.580 --> 00:55:14.650 align:middle line:90% close to that point. 00:55:14.650 --> 00:55:17.680 align:middle line:90% 00:55:17.680 --> 00:55:22.420 align:middle line:84% So now the first thing that we want to prove is-- 00:55:22.420 --> 00:55:24.980 align:middle line:84% so we want to prove that-- so we want-- 00:55:24.980 --> 00:55:28.600 align:middle line:84% our goal here is to prove this theorem. 00:55:28.600 --> 00:55:34.800 align:middle line:84% Let's say that if you have a continuous function on a closed 00:55:34.800 --> 00:55:38.520 align:middle line:84% interval-- so a closed bounded interval, compact interval, 00:55:38.520 --> 00:55:43.420 align:middle line:84% from a to b, then the function is actually Riemann integrable. 00:55:43.420 --> 00:55:46.280 align:middle line:90% 00:55:46.280 --> 00:55:49.840 align:middle line:84% And to prove that, we need again this notion 00:55:49.840 --> 00:55:52.280 align:middle line:90% of uniformly continuous. 00:55:52.280 --> 00:55:54.000 align:middle line:84% And so the first thing we want to prove 00:55:54.000 --> 00:56:00.720 align:middle line:84% is we want to upgrade continuity to uniform continuity. 00:56:00.720 --> 00:56:02.440 align:middle line:90% So what does that mean? 00:56:02.440 --> 00:56:08.280 align:middle line:84% So it means that if you take a function on a compact interval 00:56:08.280 --> 00:56:10.360 align:middle line:90% like this-- 00:56:10.360 --> 00:56:17.560 align:middle line:84% so if you have a continuous function, 00:56:17.560 --> 00:56:24.140 align:middle line:90% then f is uniformly continuous. 00:56:24.140 --> 00:56:31.780 align:middle line:90% 00:56:31.780 --> 00:56:35.540 align:middle line:84% So if the interval actually is compact, 00:56:35.540 --> 00:56:37.940 align:middle line:84% then the-- and then a continuous function 00:56:37.940 --> 00:56:39.740 align:middle line:90% is uniformly continuous. 00:56:39.740 --> 00:56:42.380 align:middle line:84% Let's just look at the two examples we had. 00:56:42.380 --> 00:56:44.180 align:middle line:90% We had two examples. 00:56:44.180 --> 00:56:49.100 align:middle line:84% In the first one, f of x was equal to x squared. 00:56:49.100 --> 00:56:51.140 align:middle line:90% But it was defined on all of-- 00:56:51.140 --> 00:56:52.980 align:middle line:90% it was continuous of course. 00:56:52.980 --> 00:56:56.520 align:middle line:84% But the thing is it wasn't on a compact interval. 00:56:56.520 --> 00:56:58.020 align:middle line:90% It was on all of R. 00:56:58.020 --> 00:57:02.260 align:middle line:84% In the second, this function here is also continuous, 00:57:02.260 --> 00:57:06.240 align:middle line:84% but it's not a compact interval, because 0 is not included. 00:57:06.240 --> 00:57:07.740 align:middle line:90% Yeah? 00:57:07.740 --> 00:57:11.900 align:middle line:84% AUDIENCE: Compact in the sense of the before? 00:57:11.900 --> 00:57:16.140 align:middle line:84% TOBIAS COLDING: So remember that-- 00:57:16.140 --> 00:57:24.300 align:middle line:84% so compact, you can either think about it as-- 00:57:24.300 --> 00:57:28.560 align:middle line:84% we talked about compact metric spaces or compact sets. 00:57:28.560 --> 00:57:35.880 align:middle line:84% But for a subset of the real line, 00:57:35.880 --> 00:57:41.720 align:middle line:84% compact is the same as saying that it's bounded and closed. 00:57:41.720 --> 00:57:46.760 align:middle line:84% And so this thing here, when I'm writing it like this, a and b 00:57:46.760 --> 00:57:48.160 align:middle line:90% are real numbers. 00:57:48.160 --> 00:57:49.620 align:middle line:90% And they are both included. 00:57:49.620 --> 00:57:54.520 align:middle line:84% And again, if it was this here, then one end point is included, 00:57:54.520 --> 00:57:55.700 align:middle line:90% but the other one is not. 00:57:55.700 --> 00:57:57.500 align:middle line:90% So it's not actually compact. 00:57:57.500 --> 00:58:00.306 align:middle line:90% 00:58:00.306 --> 00:58:04.460 align:middle line:84% And so if the function here is continuous, 00:58:04.460 --> 00:58:06.100 align:middle line:90% then it's uniformly continuous. 00:58:06.100 --> 00:58:09.120 align:middle line:90% So that's what we want to prove. 00:58:09.120 --> 00:58:16.160 align:middle line:84% Again, if the interval is compact, then actually, 00:58:16.160 --> 00:58:19.760 align:middle line:90% you get the stronger continuity. 00:58:19.760 --> 00:58:21.500 align:middle line:90% So now let's try to prove that. 00:58:21.500 --> 00:58:24.200 align:middle line:90% 00:58:24.200 --> 00:58:24.950 align:middle line:90% Suppose not. 00:58:24.950 --> 00:58:30.060 align:middle line:90% 00:58:30.060 --> 00:58:33.020 align:middle line:84% If this here was not the case, then it 00:58:33.020 --> 00:58:36.260 align:middle line:84% meant that-- so we take the negation of the definition 00:58:36.260 --> 00:58:38.180 align:middle line:90% of uniform continuity. 00:58:38.180 --> 00:58:41.460 align:middle line:84% So this means that there exists an epsilon. 00:58:41.460 --> 00:58:43.140 align:middle line:90% So suppose not. 00:58:43.140 --> 00:58:52.580 align:middle line:84% Then there exists epsilon greater than 0 00:58:52.580 --> 00:59:02.260 align:middle line:84% such that no delta would work, such that for all delta 00:59:02.260 --> 00:59:07.300 align:middle line:84% greater than 0, there exists x comma 00:59:07.300 --> 00:59:19.060 align:middle line:84% y so that the difference between these two is less than delta 00:59:19.060 --> 00:59:29.880 align:middle line:84% and the images is bigger or equal to epsilon. 00:59:29.880 --> 00:59:33.880 align:middle line:84% That's just the negation of the statement. 00:59:33.880 --> 00:59:37.380 align:middle line:84% But now we choose a sequence of delta. 00:59:37.380 --> 00:59:39.340 align:middle line:90% And so we do this-- 00:59:39.340 --> 00:59:41.120 align:middle line:90% so this is for all delta. 00:59:41.120 --> 00:59:43.800 align:middle line:84% So we should take advantage of that. 00:59:43.800 --> 00:59:48.080 align:middle line:84% We look at a sequence of such delta. 00:59:48.080 --> 00:59:51.820 align:middle line:84% So we look at a sequence of delta. 00:59:51.820 --> 01:00:03.280 align:middle line:84% And this is where we use this for delta equal to 1 over n. 01:00:03.280 --> 01:00:12.200 align:middle line:84% So n here is, again, a positive natural number, like that. 01:00:12.200 --> 01:00:21.120 align:middle line:84% So this means that we-- for each n, you have xn, and you have yn. 01:00:21.120 --> 01:00:25.020 align:middle line:84% They're both in the interval from a to b. 01:00:25.020 --> 01:00:31.120 align:middle line:84% And you have that xn minus yn is less than 1 over n. 01:00:31.120 --> 01:00:39.820 align:middle line:84% And you have that the images f of xn minus f of yn 01:00:39.820 --> 01:00:42.380 align:middle line:90% is bigger or equal to epsilon. 01:00:42.380 --> 01:00:44.960 align:middle line:84% Epsilon was positive, strictly positive. 01:00:44.960 --> 01:00:47.860 align:middle line:90% 01:00:47.860 --> 01:00:51.100 align:middle line:84% But this means-- so now we have xn. 01:00:51.100 --> 01:00:53.540 align:middle line:90% So these are in this interval. 01:00:53.540 --> 01:00:57.440 align:middle line:84% So this means that we can find a subsequence by compactness. 01:00:57.440 --> 01:01:02.620 align:middle line:84% We can find of the Bolzano-Weierstrass theorem. 01:01:02.620 --> 01:01:07.900 align:middle line:84% We have a subsequence that converges to some x. 01:01:07.900 --> 01:01:12.780 align:middle line:84% This is just a subsequence of these xn's. 01:01:12.780 --> 01:01:16.420 align:middle line:90% But now you can just look at-- 01:01:16.420 --> 01:01:19.700 align:middle line:84% so now, if you're looking at the corresponding subsequence 01:01:19.700 --> 01:01:29.120 align:middle line:84% of the yn's, so you have that yn k minus x, 01:01:29.120 --> 01:01:32.160 align:middle line:84% this thing here is less than or equal to-- 01:01:32.160 --> 01:01:35.400 align:middle line:84% by the triangle inequality, this is less than or equal 01:01:35.400 --> 01:01:46.600 align:middle line:84% to yn k minus xn k plus xn k minus x. 01:01:46.600 --> 01:01:50.240 align:middle line:90% 01:01:50.240 --> 01:01:54.840 align:middle line:84% And we have that this thing here, this here, was less than 1 01:01:54.840 --> 01:01:56.400 align:middle line:90% over n. 01:01:56.400 --> 01:02:01.320 align:middle line:90% And we have plus this here. 01:02:01.320 --> 01:02:03.680 align:middle line:84% But this thing here is actually over nk. 01:02:03.680 --> 01:02:04.455 align:middle line:90% Sorry. 01:02:04.455 --> 01:02:04.955 align:middle line:90% nk. 01:02:04.955 --> 01:02:08.560 align:middle line:90% 01:02:08.560 --> 01:02:12.080 align:middle line:84% But you have that this thing here, this here, 01:02:12.080 --> 01:02:13.940 align:middle line:90% this subsequence go to x. 01:02:13.940 --> 01:02:15.880 align:middle line:90% So this here goes to 0. 01:02:15.880 --> 01:02:18.280 align:middle line:90% And this here goes to 0. 01:02:18.280 --> 01:02:25.020 align:middle line:84% So this means that these yn k actually also converge to x. 01:02:25.020 --> 01:02:29.740 align:middle line:90% 01:02:29.740 --> 01:02:31.500 align:middle line:90% But now you have-- 01:02:31.500 --> 01:02:38.620 align:middle line:84% so you have this, and you have that yn k converge to x. 01:02:38.620 --> 01:02:42.220 align:middle line:84% So now this here together with that the function 01:02:42.220 --> 01:02:52.740 align:middle line:84% is continuous means that f xn k converges to f of x. 01:02:52.740 --> 01:03:02.660 align:middle line:84% And you have that f yn k converge to also f of x. 01:03:02.660 --> 01:03:08.800 align:middle line:84% This is just because since f is continuous. 01:03:08.800 --> 01:03:14.860 align:middle line:90% 01:03:14.860 --> 01:03:18.760 align:middle line:90% But now you just look at-- 01:03:18.760 --> 01:03:22.460 align:middle line:90% now you just look at the-- 01:03:22.460 --> 01:03:24.600 align:middle line:90% so now you have that-- 01:03:24.600 --> 01:03:31.360 align:middle line:84% so again, f xn k converges to f of x. 01:03:31.360 --> 01:03:39.360 align:middle line:84% And this here converges to f of x. 01:03:39.360 --> 01:03:46.820 align:middle line:84% But I claim that this here-- but we also have that f xn k minus f 01:03:46.820 --> 01:03:49.640 align:middle line:90% yn k-- 01:03:49.640 --> 01:03:55.920 align:middle line:84% we have that this thing here is bigger or equal to epsilon. 01:03:55.920 --> 01:03:58.680 align:middle line:84% But then you can't have-- if this here-- 01:03:58.680 --> 01:04:00.880 align:middle line:84% this here is clearly a contradiction, 01:04:00.880 --> 01:04:03.080 align:middle line:84% because if this converges to that, 01:04:03.080 --> 01:04:05.040 align:middle line:84% this here converges to the same thing. 01:04:05.040 --> 01:04:09.080 align:middle line:84% Then these two things here must get closer and closer together. 01:04:09.080 --> 01:04:12.240 align:middle line:84% But they're saying that they are some fixed thing apart. 01:04:12.240 --> 01:04:16.400 align:middle line:84% So that's clearly a contradiction. 01:04:16.400 --> 01:04:19.400 align:middle line:84% So this proof that if it is continuous, 01:04:19.400 --> 01:04:21.480 align:middle line:90% it's also uniformly continuous. 01:04:21.480 --> 01:04:24.712 align:middle line:90% 01:04:24.712 --> 01:04:32.500 align:middle line:84% So now let's use this to prove that if you take a function-- so 01:04:32.500 --> 01:04:35.980 align:middle line:84% now we are ready to get back to the theorem that 01:04:35.980 --> 01:04:39.280 align:middle line:90% was our main interest. 01:04:39.280 --> 01:04:41.420 align:middle line:84% So you see also in the two examples 01:04:41.420 --> 01:04:45.820 align:middle line:84% I had, it was no accident, again, that in the first one, 01:04:45.820 --> 01:04:47.900 align:middle line:90% the interval was unbounded. 01:04:47.900 --> 01:04:50.980 align:middle line:84% In the second one, the interval was bounded, 01:04:50.980 --> 01:04:52.800 align:middle line:90% but it was not closed. 01:04:52.800 --> 01:04:55.660 align:middle line:84% And so either case, it failed to be compact. 01:04:55.660 --> 01:04:58.460 align:middle line:84% And that was why you could have a continuous function that 01:04:58.460 --> 01:05:02.100 align:middle line:90% wasn't uniformly continuous. 01:05:02.100 --> 01:05:05.580 align:middle line:84% And so now we get back to this theorem. 01:05:05.580 --> 01:05:07.840 align:middle line:90% And so the theorem again is-- 01:05:07.840 --> 01:05:10.100 align:middle line:84% the theorem gives you this nice criterion 01:05:10.100 --> 01:05:12.960 align:middle line:84% for when a function is Riemann integrable. 01:05:12.960 --> 01:05:26.100 align:middle line:84% So if f here is continuous, then f here is Riemann integrable. 01:05:26.100 --> 01:05:29.640 align:middle line:90% 01:05:29.640 --> 01:05:31.680 align:middle line:84% And again, this here should be-- it's 01:05:31.680 --> 01:05:37.120 align:middle line:84% understood that it's f here is defined 01:05:37.120 --> 01:05:40.510 align:middle line:84% on an integral like this integral. 01:05:40.510 --> 01:05:46.264 align:middle line:90% 01:05:46.264 --> 01:05:48.420 align:middle line:84% And so now the proof is like this. 01:05:48.420 --> 01:05:55.880 align:middle line:90% 01:05:55.880 --> 01:06:06.480 align:middle line:90% We showed earlier that we only-- 01:06:06.480 --> 01:06:22.700 align:middle line:84% that we need only show that a given epsilon greater than 0, 01:06:22.700 --> 01:06:32.060 align:middle line:84% there exists a partition so that if-- 01:06:32.060 --> 01:06:42.620 align:middle line:84% such that the upper sum minus the lower sum for this partition 01:06:42.620 --> 01:06:45.160 align:middle line:84% is less than epsilon, that's all we need. 01:06:45.160 --> 01:06:48.840 align:middle line:84% We need to prove that for each epsilon, 01:06:48.840 --> 01:06:53.140 align:middle line:84% there exists a partition so that the upper and the lower sum 01:06:53.140 --> 01:06:54.820 align:middle line:84% are within epsilon of each other. 01:06:54.820 --> 01:07:04.740 align:middle line:90% 01:07:04.740 --> 01:07:07.010 align:middle line:90% So now using that-- 01:07:07.010 --> 01:07:10.300 align:middle line:90% 01:07:10.300 --> 01:07:18.510 align:middle line:84% so now-- so given epsilon greater than 0-- 01:07:18.510 --> 01:07:25.090 align:middle line:84% since f is uniformly continuous-- 01:07:25.090 --> 01:07:26.790 align:middle line:84% so actually, we're proving that f 01:07:26.790 --> 01:07:29.970 align:middle line:84% is continuous on this compact interval. 01:07:29.970 --> 01:07:31.890 align:middle line:90% It's also uniformly continuous. 01:07:31.890 --> 01:07:36.830 align:middle line:90% 01:07:36.830 --> 01:07:40.750 align:middle line:84% Since f is uniformly continuous, it 01:07:40.750 --> 01:07:48.710 align:middle line:84% means that there exists delta greater than 0 such 01:07:48.710 --> 01:08:03.790 align:middle line:84% that if x minus y is less than delta, then-- 01:08:03.790 --> 01:08:06.860 align:middle line:84% let's say delta is, let's say, less than 2 delta. 01:08:06.860 --> 01:08:13.290 align:middle line:90% 01:08:13.290 --> 01:08:23.980 align:middle line:84% Then the images-- then f of x minus f of y is now smaller 01:08:23.980 --> 01:08:24.480 align:middle line:90% than-- 01:08:24.480 --> 01:08:27.330 align:middle line:90% 01:08:27.330 --> 01:08:30.649 align:middle line:84% and then instead of epsilon, I'm using epsilon divided 01:08:30.649 --> 01:08:32.630 align:middle line:90% by the length of the interval. 01:08:32.630 --> 01:08:33.130 align:middle line:90% Sorry. 01:08:33.130 --> 01:08:34.739 align:middle line:90% And this would be strictly less. 01:08:34.739 --> 01:08:40.409 align:middle line:90% 01:08:40.409 --> 01:08:55.010 align:middle line:84% So now we let P here be a partition where 01:08:55.010 --> 01:09:03.529 align:middle line:84% each delta xi, each little interval here, 01:09:03.529 --> 01:09:07.330 align:middle line:90% is equal to delta. 01:09:07.330 --> 01:09:11.050 align:middle line:84% You just have to be, say, less than or equal to delta, 01:09:11.050 --> 01:09:12.820 align:middle line:84% but couldn't make it equal to delta. 01:09:12.820 --> 01:09:19.750 align:middle line:90% 01:09:19.750 --> 01:09:22.470 align:middle line:90% So now if this is the case-- 01:09:22.470 --> 01:09:25.470 align:middle line:90% but then we have-- 01:09:25.470 --> 01:09:28.529 align:middle line:84% so now if we're looking at one of these small intervals here-- 01:09:28.529 --> 01:09:37.630 align:middle line:90% 01:09:37.630 --> 01:09:40.670 align:middle line:84% so now, if you're looking at one of these small intervals here, 01:09:40.670 --> 01:09:43.350 align:middle line:90% here, you have xi minus 1. 01:09:43.350 --> 01:09:45.750 align:middle line:90% And here, you have xi. 01:09:45.750 --> 01:09:50.330 align:middle line:84% So if you're now looking at one of these small interval, 01:09:50.330 --> 01:09:54.630 align:middle line:84% then you have that the function is continuous here. 01:09:54.630 --> 01:10:00.950 align:middle line:84% So if you're looking at the sup of this interval-- 01:10:00.950 --> 01:10:07.170 align:middle line:84% so the sup on this interval, this 01:10:07.170 --> 01:10:11.410 align:middle line:84% is the sup of one of these intervals here, of f. 01:10:11.410 --> 01:10:15.290 align:middle line:84% The sup here, this is-- it's actually achieved 01:10:15.290 --> 01:10:16.970 align:middle line:90% by the extreme value theorem. 01:10:16.970 --> 01:10:23.610 align:middle line:84% This is achieved at some point if, say, ya-- 01:10:23.610 --> 01:10:26.436 align:middle line:90% 01:10:26.436 --> 01:10:28.650 align:middle line:90% y and maybe this-- 01:10:28.650 --> 01:10:33.570 align:middle line:90% my-- let's just say y. 01:10:33.570 --> 01:10:37.810 align:middle line:84% And likewise, on this little interval, 01:10:37.810 --> 01:10:41.940 align:middle line:90% the inf here is also achieved. 01:10:41.940 --> 01:10:45.050 align:middle line:90% 01:10:45.050 --> 01:10:47.160 align:middle line:90% And let's call that y star. 01:10:47.160 --> 01:10:51.290 align:middle line:90% 01:10:51.290 --> 01:10:53.470 align:middle line:90% So the infimum is also achieved. 01:10:53.470 --> 01:10:58.450 align:middle line:84% And that's achieved at some y star. 01:10:58.450 --> 01:11:01.110 align:middle line:84% But now this interval here have length. 01:11:01.110 --> 01:11:09.150 align:middle line:90% This here is at most delta. 01:11:09.150 --> 01:11:11.230 align:middle line:84% So this means that the difference 01:11:11.230 --> 01:11:14.320 align:middle line:84% in the value of any such two points 01:11:14.320 --> 01:11:17.850 align:middle line:84% so they are actually less-- strictly less than 2 delta. 01:11:17.850 --> 01:11:23.350 align:middle line:84% So that means that the value differ at most like this. 01:11:23.350 --> 01:11:28.150 align:middle line:84% So it means that f of y, which is actually the sup-- 01:11:28.150 --> 01:11:31.510 align:middle line:84% so this here-- so since it's a sup, 01:11:31.510 --> 01:11:32.710 align:middle line:90% I don't even need this here. 01:11:32.710 --> 01:11:33.650 align:middle line:90% It's larger than that. 01:11:33.650 --> 01:11:36.410 align:middle line:84% So I don't need to even take absolute value sign on it. 01:11:36.410 --> 01:11:42.230 align:middle line:84% But it means that the difference here is now less than epsilon 01:11:42.230 --> 01:11:43.530 align:middle line:90% over b minus a. 01:11:43.530 --> 01:11:48.038 align:middle line:90% 01:11:48.038 --> 01:11:54.710 align:middle line:84% And so this here again is M of i minus m of i, like that. 01:11:54.710 --> 01:11:57.550 align:middle line:90% 01:11:57.550 --> 01:12:01.850 align:middle line:84% And so now you see that-- so now let's look at the contribution. 01:12:01.850 --> 01:12:08.530 align:middle line:84% So for the upper sum, we have that this here is this capital 01:12:08.530 --> 01:12:11.850 align:middle line:90% Mi and then delta xi. 01:12:11.850 --> 01:12:18.890 align:middle line:84% For the lower sum, we have that this is the sum of the small mi 01:12:18.890 --> 01:12:19.810 align:middle line:90% delta xi. 01:12:19.810 --> 01:12:22.330 align:middle line:90% 01:12:22.330 --> 01:12:26.890 align:middle line:84% And so now let's look at the difference between these two. 01:12:26.890 --> 01:12:31.450 align:middle line:84% That's the one we need to prove is strictly less than epsilon. 01:12:31.450 --> 01:12:40.850 align:middle line:84% So if you're looking at this U f of P minus L f P, 01:12:40.850 --> 01:12:43.650 align:middle line:90% this here is now the sum here. 01:12:43.650 --> 01:12:46.530 align:middle line:84% And then it's-- so you're summing-- 01:12:46.530 --> 01:12:47.830 align:middle line:90% this thing is the same. 01:12:47.830 --> 01:12:50.690 align:middle line:84% So you can write the sum here-- the difference. 01:12:50.690 --> 01:12:53.090 align:middle line:90% Well, let's just write it first. 01:12:53.090 --> 01:12:55.730 align:middle line:90% Just write it out like this. 01:12:55.730 --> 01:12:57.090 align:middle line:90% But then now I can-- 01:12:57.090 --> 01:12:59.450 align:middle line:84% once I have written it out like this, 01:12:59.450 --> 01:13:05.230 align:middle line:84% see that I can write this thing here is the sum of capital Mi 01:13:05.230 --> 01:13:09.070 align:middle line:90% minus little mi times delta xi. 01:13:09.070 --> 01:13:12.470 align:middle line:90% 01:13:12.470 --> 01:13:15.350 align:middle line:84% And this thing here I already figured-- 01:13:15.350 --> 01:13:16.730 align:middle line:90% I already have a bound. 01:13:16.730 --> 01:13:19.350 align:middle line:84% So this is each of these is less than this. 01:13:19.350 --> 01:13:21.290 align:middle line:90% So I just write in that bound. 01:13:21.290 --> 01:13:24.470 align:middle line:90% So this here is less than-- 01:13:24.470 --> 01:13:29.110 align:middle line:84% and then the sum here, epsilon over b minus a-- 01:13:29.110 --> 01:13:32.990 align:middle line:90% and then like this. 01:13:32.990 --> 01:13:34.770 align:middle line:90% But now this is just a constant. 01:13:34.770 --> 01:13:37.820 align:middle line:84% So this just-- I can just take this out of the summation sign. 01:13:37.820 --> 01:13:41.230 align:middle line:90% 01:13:41.230 --> 01:13:42.990 align:middle line:90% And so I get this thing. 01:13:42.990 --> 01:13:45.570 align:middle line:84% So this is the sum of the small interval. 01:13:45.570 --> 01:13:49.070 align:middle line:84% But the sum of the small interval is the whole interval. 01:13:49.070 --> 01:13:54.890 align:middle line:84% So this is epsilon b minus a times b minus a. 01:13:54.890 --> 01:13:58.030 align:middle line:90% And so this here is epsilon. 01:13:58.030 --> 01:13:59.950 align:middle line:90% And so you have exactly-- 01:13:59.950 --> 01:14:06.510 align:middle line:84% it's a-- this difference here is strictly bounded by epsilon. 01:14:06.510 --> 01:14:13.230 align:middle line:84% So let's prove that if the function is continuous, 01:14:13.230 --> 01:14:16.090 align:middle line:84% then it is on a compact interval. 01:14:16.090 --> 01:14:19.090 align:middle line:90% Then it is Riemann integral. 01:14:19.090 --> 01:14:23.330 align:middle line:84% And now let me at least state some of the basic formulas 01:14:23.330 --> 01:14:25.990 align:middle line:90% for Riemann integrals. 01:14:25.990 --> 01:14:29.410 align:middle line:90% 01:14:29.410 --> 01:14:33.320 align:middle line:84% And so maybe we might prove some of these next time. 01:14:33.320 --> 01:14:42.610 align:middle line:90% 01:14:42.610 --> 01:14:47.190 align:middle line:84% So the first one is that if you take-- 01:14:47.190 --> 01:14:49.670 align:middle line:90% so you have this theorem here. 01:14:49.670 --> 01:14:52.970 align:middle line:90% 01:14:52.970 --> 01:14:54.770 align:middle line:84% The first one is-- so suppose you 01:14:54.770 --> 01:14:58.450 align:middle line:84% take a function that is Riemann integrable 01:14:58.450 --> 01:15:01.910 align:middle line:90% on some integral like this. 01:15:01.910 --> 01:15:04.630 align:middle line:90% And c here is constant. 01:15:04.630 --> 01:15:07.350 align:middle line:90% It's a real number. 01:15:07.350 --> 01:15:13.210 align:middle line:84% Then this constant times f is also Riemann integrable. 01:15:13.210 --> 01:15:16.910 align:middle line:90% 01:15:16.910 --> 01:15:20.470 align:middle line:84% And the integral of this function 01:15:20.470 --> 01:15:29.310 align:middle line:84% here is equal to c times the integral, Riemann integral, 01:15:29.310 --> 01:15:30.670 align:middle line:90% like that. 01:15:30.670 --> 01:15:31.970 align:middle line:90% So that's the first one. 01:15:31.970 --> 01:15:34.950 align:middle line:90% 01:15:34.950 --> 01:15:45.270 align:middle line:84% The next one is that if you take two functions, f and g. 01:15:45.270 --> 01:15:52.390 align:middle line:84% Suppose you take two functions, f and g, 01:15:52.390 --> 01:15:53.520 align:middle line:90% that are Riemann integral. 01:15:53.520 --> 01:15:59.090 align:middle line:90% 01:15:59.090 --> 01:16:02.770 align:middle line:84% And now let's look at the function that 01:16:02.770 --> 01:16:05.090 align:middle line:90% is the sum of these two. 01:16:05.090 --> 01:16:07.050 align:middle line:90% So this is a new function. 01:16:07.050 --> 01:16:10.870 align:middle line:84% And the claim is that this here is also Riemann integrable. 01:16:10.870 --> 01:16:15.170 align:middle line:90% 01:16:15.170 --> 01:16:26.910 align:middle line:84% And the integral of the sum is the sum of the integrals. 01:16:26.910 --> 01:16:27.410 align:middle line:90% Oops. 01:16:27.410 --> 01:16:37.330 align:middle line:90% 01:16:37.330 --> 01:16:40.650 align:middle line:90% That's the second one. 01:16:40.650 --> 01:16:43.930 align:middle line:90% And the third one is that-- 01:16:43.930 --> 01:16:51.130 align:middle line:84% so the third one is that if you take a function-- two functions, 01:16:51.130 --> 01:16:53.630 align:middle line:84% again, they are both Riemann integrable. 01:16:53.630 --> 01:16:56.150 align:middle line:90% 01:16:56.150 --> 01:17:00.590 align:middle line:84% And you have that f here is less than or equal to g. 01:17:00.590 --> 01:17:05.710 align:middle line:84% Then the Riemann integral of f is 01:17:05.710 --> 01:17:12.470 align:middle line:84% less than or equal to the Riemann integral of g, 01:17:12.470 --> 01:17:13.090 align:middle line:90% like that. 01:17:13.090 --> 01:17:15.790 align:middle line:90% 01:17:15.790 --> 01:17:21.710 align:middle line:90% And the fourth rule is that-- 01:17:21.710 --> 01:17:23.990 align:middle line:84% the fourth rule is a little bit different. 01:17:23.990 --> 01:17:28.170 align:middle line:84% So this is if you split the integral. 01:17:28.170 --> 01:17:31.394 align:middle line:90% 01:17:31.394 --> 01:17:35.810 align:middle line:90% So the fourth rule is like this. 01:17:35.810 --> 01:17:38.650 align:middle line:84% Again, I will prove some of them, not all of them, 01:17:38.650 --> 01:17:42.490 align:middle line:84% but I'll prove some of them next time but not all of them. 01:17:42.490 --> 01:17:45.190 align:middle line:84% I won't prove the first one for sure. 01:17:45.190 --> 01:17:48.610 align:middle line:84% So the fourth is that if you take-- 01:17:48.610 --> 01:17:53.670 align:middle line:84% so imagine-- so you have this integral here from a to b. 01:17:53.670 --> 01:17:56.902 align:middle line:84% But imagine that-- so here, you have a. 01:17:56.902 --> 01:17:59.010 align:middle line:90% Here, you have b. 01:17:59.010 --> 01:18:04.650 align:middle line:84% Imagine that c is somewhere in between a and b. 01:18:04.650 --> 01:18:08.450 align:middle line:84% And suppose that you have a function that is Riemann 01:18:08.450 --> 01:18:13.010 align:middle line:90% integrable on this interval. 01:18:13.010 --> 01:18:18.010 align:middle line:84% Then the claim is that then, if you take the function 01:18:18.010 --> 01:18:23.730 align:middle line:84% and you restrict it to either of this interval or that interval, 01:18:23.730 --> 01:18:25.510 align:middle line:90% it's still Riemann integrable. 01:18:25.510 --> 01:18:31.610 align:middle line:90% 01:18:31.610 --> 01:18:36.050 align:middle line:90% And f here is also in-- 01:18:36.050 --> 01:18:37.350 align:middle line:90% from c to b. 01:18:37.350 --> 01:18:40.130 align:middle line:90% 01:18:40.130 --> 01:18:45.330 align:middle line:84% And the claim is that the Riemann integral 01:18:45.330 --> 01:18:56.790 align:middle line:84% on the original integral is the sum of the Riemann integral, 01:18:56.790 --> 01:18:58.775 align:middle line:90% like that. 01:18:58.775 --> 01:19:03.390 align:middle line:84% And so if you think about it in terms of area below the graph, 01:19:03.390 --> 01:19:06.510 align:middle line:84% if you take a function like this, 01:19:06.510 --> 01:19:10.270 align:middle line:84% if this here was-- if this here was c, then 01:19:10.270 --> 01:19:14.150 align:middle line:84% the area below the graph here, I guess 01:19:14.150 --> 01:19:16.490 align:middle line:84% this here would be sign area below the graph. 01:19:16.490 --> 01:19:19.190 align:middle line:84% But if you just took the area below the function 01:19:19.190 --> 01:19:22.390 align:middle line:84% was actually positive and you think about the Riemann 01:19:22.390 --> 01:19:26.565 align:middle line:84% integral as the area below the graph, if this here is c, 01:19:26.565 --> 01:19:31.990 align:middle line:84% this here is a, this here is b, then this area here 01:19:31.990 --> 01:19:33.990 align:middle line:90% plus this area is-- 01:19:33.990 --> 01:19:38.370 align:middle line:84% the sum of these two is the sum underneath the whole graph. 01:19:38.370 --> 01:19:40.020 align:middle line:90% So it makes sense. 01:19:40.020 --> 01:19:43.430 align:middle line:90% 01:19:43.430 --> 01:19:44.610 align:middle line:90% Any questions? 01:19:44.610 --> 01:19:47.190 align:middle line:90% 01:19:47.190 --> 01:19:47.850 align:middle line:90% OK, great. 01:19:47.850 --> 01:19:49.980 align:middle line:90% See you next time. 01:19:49.980 --> 01:19:52.000 align:middle line:90%