WEBVTT 00:00:00.000 --> 00:00:02.485 align:middle line:90% [SQUEAKING] 00:00:02.485 --> 00:00:03.976 align:middle line:90% [RUSTLING] 00:00:03.976 --> 00:00:12.665 align:middle line:90% [CLICKING] 00:00:12.665 --> 00:00:14.040 align:middle line:84% CASEY RODRIGUEZ: Last lecture, we 00:00:14.040 --> 00:00:16.500 align:middle line:84% introduced the notion of the limit of a function 00:00:16.500 --> 00:00:21.210 align:middle line:84% as x goes to c, which we write limit x arrow 00:00:21.210 --> 00:00:23.800 align:middle line:84% c, f of x equals L. What does this mean? 00:00:23.800 --> 00:00:25.470 align:middle line:84% This means for all epsilon positive, 00:00:25.470 --> 00:00:28.920 align:middle line:84% there exists a delta positive such that for all x in S 00:00:28.920 --> 00:00:32.790 align:middle line:84% satisfying 0 is less than x minus c is less than delta, 00:00:32.790 --> 00:00:35.505 align:middle line:84% we have that f of x minus L is less than epsilon. 00:00:35.505 --> 00:00:38.190 align:middle line:90% 00:00:38.190 --> 00:00:41.430 align:middle line:84% And we proved the following theorem last time, 00:00:41.430 --> 00:00:46.850 align:middle line:84% that if we have a set, a cluster point of S-- 00:00:46.850 --> 00:00:50.690 align:middle line:84% so this is where we look at limits and f 00:00:50.690 --> 00:00:54.290 align:middle line:84% as a function from S to R-- then the limit as x 00:00:54.290 --> 00:00:59.810 align:middle line:84% goes to c of f of x equals L if and only if for every sequence 00:00:59.810 --> 00:01:05.780 align:middle line:84% x sub n converging to c, we have f of x n, the new sequence, 00:01:05.780 --> 00:01:11.090 align:middle line:84% converges to L. So this theorem here connects limits 00:01:11.090 --> 00:01:15.200 align:middle line:84% of functions to what we did previously, 00:01:15.200 --> 00:01:18.930 align:middle line:90% limits of sequences. 00:01:18.930 --> 00:01:20.760 align:middle line:84% Now, using this theorem, we'll get 00:01:20.760 --> 00:01:29.250 align:middle line:84% analogs of theorems we proved for sequences, but now 00:01:29.250 --> 00:01:30.255 align:middle line:90% for limits. 00:01:30.255 --> 00:01:34.490 align:middle line:90% 00:01:34.490 --> 00:01:37.475 align:middle line:84% First, let me show a few simple applications of this theorem. 00:01:37.475 --> 00:01:44.030 align:middle line:90% 00:01:44.030 --> 00:01:45.970 align:middle line:84% So for example, we could prove the following-- 00:01:45.970 --> 00:01:55.120 align:middle line:84% that for all c in R, the limit as x goes to c of x squared 00:01:55.120 --> 00:01:57.010 align:middle line:90% equals c squared. 00:01:57.010 --> 00:02:02.505 align:middle line:84% So we could have done this using just the definition, 00:02:02.505 --> 00:02:03.880 align:middle line:84% but now with this theorem, we can 00:02:03.880 --> 00:02:08.620 align:middle line:84% prove it a little bit quicker and easier, 00:02:08.620 --> 00:02:12.970 align:middle line:84% because we essentially did all the hard work when 00:02:12.970 --> 00:02:16.270 align:middle line:84% we proved that the product of two convergent sequences 00:02:16.270 --> 00:02:20.090 align:middle line:84% is convergent, which is what we'll use here. 00:02:20.090 --> 00:02:23.800 align:middle line:84% So we're going to prove this theorem using 00:02:23.800 --> 00:02:25.750 align:middle line:90% the previous theorem. 00:02:25.750 --> 00:02:30.180 align:middle line:90% So let x n-- 00:02:30.180 --> 00:02:33.660 align:middle line:84% so for this theorem, since it's not stated, 00:02:33.660 --> 00:02:39.050 align:middle line:84% you should take the function to be f of x equals x squared. 00:02:39.050 --> 00:02:41.870 align:middle line:84% And then the set that is defined on S is equal to R 00:02:41.870 --> 00:02:51.560 align:middle line:84% So let x n be a sequence such that x n converges to c as n 00:02:51.560 --> 00:02:54.530 align:middle line:90% goes to infinity. 00:02:54.530 --> 00:02:57.530 align:middle line:84% We now want to show that f of x n 00:02:57.530 --> 00:03:01.410 align:middle line:84% converges to f of c, where f of x equals x squared. 00:03:01.410 --> 00:03:04.400 align:middle line:84% But this follows from what we proved for limits. 00:03:04.400 --> 00:03:10.700 align:middle line:84% So we proved the product of two convergent sequences 00:03:10.700 --> 00:03:11.540 align:middle line:90% is convergent. 00:03:11.540 --> 00:03:15.180 align:middle line:90% 00:03:15.180 --> 00:03:25.770 align:middle line:84% By earlier theorem about the product 00:03:25.770 --> 00:03:37.470 align:middle line:84% of convergent sequences, we get that x n squared converges 00:03:37.470 --> 00:03:44.280 align:middle line:90% to c squared, which implies-- 00:03:44.280 --> 00:03:47.160 align:middle line:84% so we've now verified for every sequence converging 00:03:47.160 --> 00:03:51.270 align:middle line:84% to c, f of x n, meaning x n squared, 00:03:51.270 --> 00:03:56.070 align:middle line:90% converges to c squared. 00:03:56.070 --> 00:04:02.030 align:middle line:84% And therefore by the previous theorem, 00:04:02.030 --> 00:04:03.440 align:middle line:90% we've proven the claim. 00:04:03.440 --> 00:04:16.170 align:middle line:90% 00:04:16.170 --> 00:04:20.415 align:middle line:84% So now let's use this term to study a couple of more limits. 00:04:20.415 --> 00:04:23.460 align:middle line:90% 00:04:23.460 --> 00:04:25.100 align:middle line:84% And we use this theorem, in fact, 00:04:25.100 --> 00:04:28.870 align:middle line:84% to show that a certain limit does not exist. 00:04:28.870 --> 00:04:35.700 align:middle line:84% So the limit as x goes to 0 of sine 1 over x-- 00:04:35.700 --> 00:04:38.820 align:middle line:90% 00:04:38.820 --> 00:04:45.860 align:middle line:90% this limit does not exist. 00:04:45.860 --> 00:04:50.080 align:middle line:84% So remember for a limit, I look at all those points 00:04:50.080 --> 00:04:53.650 align:middle line:84% x that are close to c, but not equal to c. 00:04:53.650 --> 00:04:57.310 align:middle line:84% So this function, which I'm taking the limit 00:04:57.310 --> 00:05:01.630 align:middle line:84% as x goes to 0 of, doesn't need to be 00:05:01.630 --> 00:05:05.680 align:middle line:84% defined at x equals 0 in order to consider the limit. 00:05:05.680 --> 00:05:08.950 align:middle line:84% And if you like, the function sine 00:05:08.950 --> 00:05:17.080 align:middle line:84% of 1 over x is defined on S equals R take away 0. 00:05:17.080 --> 00:05:20.950 align:middle line:84% So this limit does not exist, but the limit 00:05:20.950 --> 00:05:26.320 align:middle line:84% as x goes to 0 of x times sine of 1 over x does 00:05:26.320 --> 00:05:29.020 align:middle line:90% exist and equals 0. 00:05:29.020 --> 00:05:32.110 align:middle line:90% 00:05:32.110 --> 00:05:34.270 align:middle line:84% Now I want you to note something-- 00:05:34.270 --> 00:05:38.860 align:middle line:84% you can't just stick in x equals 0 here to evaluate the limit. 00:05:38.860 --> 00:05:42.490 align:middle line:84% Because then you'll be taking sine of 1 over 0 times 0. 00:05:42.490 --> 00:05:48.060 align:middle line:84% You can't divide by 0, so can't just say this limit 00:05:48.060 --> 00:05:50.190 align:middle line:84% equals 0 because I stick in x equals 0. 00:05:50.190 --> 00:05:53.040 align:middle line:84% In fact, we saw in the previous lecture 00:05:53.040 --> 00:05:59.160 align:middle line:84% that the limit need not equal the function evaluated 00:05:59.160 --> 00:06:00.490 align:middle line:90% at that point. 00:06:00.490 --> 00:06:05.850 align:middle line:84% So just to recall the example from last time, 00:06:05.850 --> 00:06:12.540 align:middle line:84% when I had f of x equals 1 if x equals 0 and 2 if x does not 00:06:12.540 --> 00:06:17.850 align:middle line:84% equal to 0, then we showed that limit 00:06:17.850 --> 00:06:22.380 align:middle line:84% as x goes to 0 of f of x equals 2, which note, 00:06:22.380 --> 00:06:25.200 align:middle line:90% does not equal f of 0. 00:06:25.200 --> 00:06:27.360 align:middle line:84% When we discuss continuity, that's 00:06:27.360 --> 00:06:32.430 align:middle line:84% what connects the limit to the function evaluated 00:06:32.430 --> 00:06:33.553 align:middle line:90% at that point. 00:06:33.553 --> 00:06:35.220 align:middle line:84% I just want to make that little comment. 00:06:35.220 --> 00:06:38.020 align:middle line:90% 00:06:38.020 --> 00:06:41.135 align:middle line:84% To prove this theorem, we're going to use, again, 00:06:41.135 --> 00:06:42.010 align:middle line:90% the previous theorem. 00:06:42.010 --> 00:06:45.610 align:middle line:84% Maybe I should-- no, I'm not going to label it. 00:06:45.610 --> 00:06:47.290 align:middle line:84% You'll know when I point up to it 00:06:47.290 --> 00:06:49.960 align:middle line:84% that I'm referring to that theorem that I stated up there. 00:06:49.960 --> 00:06:54.160 align:middle line:90% 00:06:54.160 --> 00:07:01.280 align:middle line:84% So in fact, let's prove the second limit 00:07:01.280 --> 00:07:04.900 align:middle line:90% exists and equals 0 first. 00:07:04.900 --> 00:07:10.960 align:middle line:84% So suppose x n is a sequence converging to 0. 00:07:10.960 --> 00:07:24.087 align:middle line:84% We now want to show that x sub n times sine of 1 00:07:24.087 --> 00:07:26.990 align:middle line:90% over x sub n converges to 0. 00:07:26.990 --> 00:07:29.570 align:middle line:84% And then the previous theorem up there 00:07:29.570 --> 00:07:32.360 align:middle line:84% will imply that the limit as x goes 00:07:32.360 --> 00:07:34.940 align:middle line:84% to 0 of x times sine of 1 over x equals 0. 00:07:34.940 --> 00:07:38.850 align:middle line:90% 00:07:38.850 --> 00:07:44.100 align:middle line:84% Now, if I look at the absolute value of x sub n times 00:07:44.100 --> 00:07:49.230 align:middle line:84% sine of 1 over x sub n, this is equal to the absolute value 00:07:49.230 --> 00:07:52.070 align:middle line:84% of x sub n times the absolute value of sine of 1 00:07:52.070 --> 00:07:52.700 align:middle line:90% over x sub n. 00:07:52.700 --> 00:07:57.630 align:middle line:90% 00:07:57.630 --> 00:08:00.600 align:middle line:84% And no matter what you stick into sine, 00:08:00.600 --> 00:08:03.180 align:middle line:84% sine is always bounded between 1 and minus 1. 00:08:03.180 --> 00:08:05.460 align:middle line:84% So the absolute value is always bounded by 1. 00:08:05.460 --> 00:08:11.810 align:middle line:90% 00:08:11.810 --> 00:08:14.960 align:middle line:84% So just to summarize, we've shown that-- 00:08:14.960 --> 00:08:22.430 align:middle line:90% 00:08:22.430 --> 00:08:24.660 align:middle line:84% and now we apply the squeeze theorem. 00:08:24.660 --> 00:08:27.860 align:middle line:90% So that goes to 0. 00:08:27.860 --> 00:08:30.410 align:middle line:84% It's just the constant sequence equal to 0. 00:08:30.410 --> 00:08:32.630 align:middle line:84% Since x sub n is converging to 0-- 00:08:32.630 --> 00:08:35.059 align:middle line:90% remember we're assuming that-- 00:08:35.059 --> 00:08:36.799 align:middle line:90% this converges to 0. 00:08:36.799 --> 00:08:40.699 align:middle line:84% And therefore, what gets trapped in between goes to 0. 00:08:40.699 --> 00:08:43.360 align:middle line:90% 00:08:43.360 --> 00:08:49.707 align:middle line:84% So by squeeze theorem, x sub n times sine of 1 00:08:49.707 --> 00:08:52.610 align:middle line:90% over x sub n converges to 0. 00:08:52.610 --> 00:09:11.760 align:middle line:90% And that proves the second claim 00:09:11.760 --> 00:09:15.680 align:middle line:90% So now we'll prove 1. 00:09:15.680 --> 00:09:21.840 align:middle line:84% And in the previous lecture, I did negate this definition. 00:09:21.840 --> 00:09:25.220 align:middle line:84% So let's actually negate this theorem, 00:09:25.220 --> 00:09:32.780 align:middle line:84% if you like, or use the negation of each side of this if 00:09:32.780 --> 00:09:37.010 align:middle line:84% and only if to state an equivalent theorem. 00:09:37.010 --> 00:09:42.580 align:middle line:90% 00:09:42.580 --> 00:09:44.450 align:middle line:84% So two statements are equivalent, 00:09:44.450 --> 00:09:46.960 align:middle line:84% which is in that theorem, if and only if their negations 00:09:46.960 --> 00:09:47.950 align:middle line:90% are also equivalent. 00:09:47.950 --> 00:09:53.980 align:middle line:84% So by the theorem, we also have the following fact-- 00:09:53.980 --> 00:09:59.660 align:middle line:84% limit x goes to c of f of x is not equal L if 00:09:59.660 --> 00:10:03.307 align:middle line:90% and only if there exist-- 00:10:03.307 --> 00:10:05.890 align:middle line:84% so negating the right-hand side of that if and only if-- there 00:10:05.890 --> 00:10:10.045 align:middle line:84% exists a sequence x n converging to c-- 00:10:10.045 --> 00:10:12.580 align:middle line:90% 00:10:12.580 --> 00:10:19.270 align:middle line:90% so this sequence will-- 00:10:19.270 --> 00:10:23.730 align:middle line:90% 00:10:23.730 --> 00:10:25.560 align:middle line:84% consisting of elements of S take away 00:10:25.560 --> 00:10:31.710 align:middle line:84% c, such that x n converges to c, but we 00:10:31.710 --> 00:10:47.580 align:middle line:84% don't have f of x n converging to L. And when I write this, 00:10:47.580 --> 00:10:51.030 align:middle line:84% you should read this as either this limit exists and does not 00:10:51.030 --> 00:10:52.875 align:middle line:84% equal L or this limit does not exist. 00:10:52.875 --> 00:10:59.300 align:middle line:90% 00:10:59.300 --> 00:11:02.820 align:middle line:90% So I'll even write that out-- 00:11:02.820 --> 00:11:22.660 align:middle line:84% and either limit does not exist or does not 00:11:22.660 --> 00:11:27.340 align:middle line:84% equal L. So again, that's an equivalent 00:11:27.340 --> 00:11:31.420 align:middle line:84% way of stating the theorem up there 00:11:31.420 --> 00:11:32.740 align:middle line:90% is in terms of the negation. 00:11:32.740 --> 00:11:38.260 align:middle line:84% Two statements are equivalent, which is right there, if 00:11:38.260 --> 00:11:40.450 align:middle line:84% and only if their negations are equivalent. 00:11:40.450 --> 00:11:51.550 align:middle line:90% 00:11:51.550 --> 00:11:56.640 align:middle line:84% So now, we're going to prove that the limit as x goes 00:11:56.640 --> 00:11:58.470 align:middle line:84% to 0 of sine of 1 over x does not 00:11:58.470 --> 00:12:01.680 align:middle line:84% exist by showing that there exists a sequence converging 00:12:01.680 --> 00:12:07.080 align:middle line:84% to 0 such that when I plug that in to sine, 00:12:07.080 --> 00:12:09.015 align:middle line:84% the limit of that sequence does not exist. 00:12:09.015 --> 00:12:11.850 align:middle line:90% 00:12:11.850 --> 00:12:21.740 align:middle line:84% So to show there exists a sequence converging 00:12:21.740 --> 00:12:37.120 align:middle line:84% to 0 such that limit as n goes to infinity of sine of 1 00:12:37.120 --> 00:12:43.190 align:middle line:90% over x sub n does not exist. 00:12:43.190 --> 00:12:49.940 align:middle line:84% Now, sine oscillates between 1 and minus 1 00:12:49.940 --> 00:12:54.080 align:middle line:84% depending on a certain multiple of pi over 2. 00:12:54.080 --> 00:12:58.110 align:middle line:90% 00:12:58.110 --> 00:13:02.640 align:middle line:84% So here's the intuition behind it, 00:13:02.640 --> 00:13:05.410 align:middle line:84% behind the sequence I'm about to give you-- 00:13:05.410 --> 00:13:23.440 align:middle line:84% note that sine of x equals 1 if x equals pi over 2, 00:13:23.440 --> 00:13:37.890 align:middle line:84% 5 pi over 2, 9 pi over 2, and so on, minus 1 if x 00:13:37.890 --> 00:13:46.770 align:middle line:84% equals 3 pi over 2, 7 pi over 2, 11 pi over 2, and so on. 00:13:46.770 --> 00:13:51.110 align:middle line:84% So I can stick in things that are getting bigger 00:13:51.110 --> 00:13:53.510 align:middle line:90% to sine and get 1 or minus 1. 00:13:53.510 --> 00:13:56.450 align:middle line:84% And in fact, let me change this to y, 00:13:56.450 --> 00:14:03.900 align:middle line:84% since we're using x to be essentially 1 over y. 00:14:03.900 --> 00:14:08.040 align:middle line:84% So if I stick in pi over 2, 5 pi over 2, 9 pi over 2, and so on, 00:14:08.040 --> 00:14:09.630 align:middle line:90% I get sine equals 1. 00:14:09.630 --> 00:14:13.050 align:middle line:84% If it's 3 pi over 2, 7 pi over 2, 11 pi over 2, 00:14:13.050 --> 00:14:16.530 align:middle line:84% and I stick that into sine, I get minus 1. 00:14:16.530 --> 00:14:21.120 align:middle line:84% But that means if I stick 1 over these numbers 00:14:21.120 --> 00:14:27.330 align:middle line:84% into sine of 1 over x, I get 1 or minus 1. 00:14:27.330 --> 00:14:29.940 align:middle line:84% And so then that sequence that I get 00:14:29.940 --> 00:14:32.910 align:middle line:84% will be 1 or minus 1 alternating. 00:14:32.910 --> 00:14:35.100 align:middle line:84% And we know that sequence does not converge. 00:14:35.100 --> 00:14:36.550 align:middle line:90% So that's the idea. 00:14:36.550 --> 00:14:38.220 align:middle line:90% So let me write that down. 00:14:38.220 --> 00:14:44.740 align:middle line:90% 00:14:44.740 --> 00:14:50.970 align:middle line:84% Let x sub n be 1 over these numbers, essentially, 00:14:50.970 --> 00:15:01.330 align:middle line:84% so 2n minus 1, pi over 2, minus 1, 00:15:01.330 --> 00:15:03.557 align:middle line:84% because we're going to stick this into sine of 1 00:15:03.557 --> 00:15:10.150 align:middle line:84% over x, which is 2 over 2n minus 1 over pi. 00:15:10.150 --> 00:15:17.920 align:middle line:90% 00:15:17.920 --> 00:15:29.630 align:middle line:84% Now, note for all n, x sub n is less than or equal to-- 00:15:29.630 --> 00:15:35.690 align:middle line:84% I can write this as 2 over n plus n minus 1 pi. 00:15:35.690 --> 00:15:37.850 align:middle line:84% And since n is bigger than or equal to 1, 00:15:37.850 --> 00:15:39.740 align:middle line:84% this is always bigger than or equal to 0. 00:15:39.740 --> 00:15:43.700 align:middle line:84% So this is less than or equal to 2 over n pi. 00:15:43.700 --> 00:15:44.870 align:middle line:90% And this goes to 0. 00:15:44.870 --> 00:15:51.640 align:middle line:84% So that shows by the squeeze theorem 00:15:51.640 --> 00:15:56.930 align:middle line:84% that this sequence I've defined here converges to 0. 00:15:56.930 --> 00:15:59.060 align:middle line:84% But what happens when I plug this sequence 00:15:59.060 --> 00:16:01.370 align:middle line:90% into sine of 1 over x? 00:16:01.370 --> 00:16:10.840 align:middle line:90% 00:16:10.840 --> 00:16:18.595 align:middle line:84% This is now equal to sine of 2n minus 1 pi over 2. 00:16:18.595 --> 00:16:21.900 align:middle line:90% 00:16:21.900 --> 00:16:26.150 align:middle line:84% And this is therefore equal to 1 minus 1, 00:16:26.150 --> 00:16:29.790 align:middle line:90% 1, minus 1, 1, minus 1. 00:16:29.790 --> 00:16:40.530 align:middle line:90% 00:16:40.530 --> 00:16:43.230 align:middle line:84% And this sequence, which is just equal to minus 1 00:16:43.230 --> 00:16:46.545 align:middle line:90% to the n plus 1-- 00:16:46.545 --> 00:16:51.490 align:middle line:84% I don't know why I capitalized that-- does not converge. 00:16:51.490 --> 00:16:56.050 align:middle line:90% 00:16:56.050 --> 00:16:59.350 align:middle line:84% So we found a sequence converging to 0 such 00:16:59.350 --> 00:17:01.570 align:middle line:84% that when I stick it into the function, 00:17:01.570 --> 00:17:04.180 align:middle line:84% that new sequence does not converge. 00:17:04.180 --> 00:17:12.570 align:middle line:84% So we've proven that this limit does not exist. 00:17:12.570 --> 00:17:30.290 align:middle line:90% 00:17:30.290 --> 00:17:41.506 align:middle line:84% So I alluded to this fact that this theorem will give us 00:17:41.506 --> 00:17:43.380 align:middle line:84% theorems that are similar to what 00:17:43.380 --> 00:17:48.340 align:middle line:84% we proved for sequences, except now for limits of functions. 00:17:48.340 --> 00:17:53.220 align:middle line:84% So let me just state the simplest theorem you can get. 00:17:53.220 --> 00:17:56.510 align:middle line:90% 00:17:56.510 --> 00:18:14.400 align:middle line:84% So let S be a subset of R, c a cluster point of S, 00:18:14.400 --> 00:18:24.570 align:middle line:84% and suppose I have two functions f going from S to R 00:18:24.570 --> 00:18:33.710 align:middle line:90% and g going from S to R. 00:18:33.710 --> 00:18:52.860 align:middle line:84% If these two limits exist, and one function 00:18:52.860 --> 00:19:01.050 align:middle line:84% is smaller than the other, then-- 00:19:01.050 --> 00:19:05.700 align:middle line:84% so we had an analogous statement for sequences, 00:19:05.700 --> 00:19:08.280 align:middle line:84% which was if I have two sequences converging 00:19:08.280 --> 00:19:10.410 align:middle line:84% and one is less than or equal to the other, 00:19:10.410 --> 00:19:16.360 align:middle line:84% then one limit is less than or equal to the other. 00:19:16.360 --> 00:19:20.640 align:middle line:84% And it's an analogous conclusion for limits of functions. 00:19:20.640 --> 00:19:33.070 align:middle line:90% 00:19:33.070 --> 00:19:36.880 align:middle line:84% So again, the analogous statement for sequences 00:19:36.880 --> 00:19:39.190 align:middle line:84% was we have two sequences, one less than 00:19:39.190 --> 00:19:42.700 align:middle line:84% or equal to the other, which then the limit of the smaller 00:19:42.700 --> 00:19:45.200 align:middle line:84% sequence is less than or equal to the limit of the bigger 00:19:45.200 --> 00:19:45.700 align:middle line:90% sequence. 00:19:45.700 --> 00:19:54.550 align:middle line:90% 00:19:54.550 --> 00:19:56.080 align:middle line:90% So let's give the proof. 00:19:56.080 --> 00:20:00.850 align:middle line:84% And we'll use this theorem connecting limits of functions 00:20:00.850 --> 00:20:02.260 align:middle line:90% to limits of sequences. 00:20:02.260 --> 00:20:04.360 align:middle line:84% And then we'll use the corresponding statement 00:20:04.360 --> 00:20:07.090 align:middle line:84% which we do have for sequences, which I just 00:20:07.090 --> 00:20:08.740 align:middle line:90% stated a couple of times. 00:20:08.740 --> 00:20:16.100 align:middle line:84% So let L1 be the limit as x goes to c 00:20:16.100 --> 00:20:27.160 align:middle line:84% of f of x, and L2 the limit as x goes to c of g of x. 00:20:27.160 --> 00:20:30.025 align:middle line:84% And what we want to show is that L1 is less than or equal to L2. 00:20:30.025 --> 00:20:44.220 align:middle line:90% 00:20:44.220 --> 00:20:53.650 align:middle line:84% Let x n be the sequence, and S takeaway c such 00:20:53.650 --> 00:20:57.120 align:middle line:90% that x n converges to c. 00:20:57.120 --> 00:21:02.910 align:middle line:84% Such a sequence exists because c is a cluster point of s. 00:21:02.910 --> 00:21:05.460 align:middle line:84% And you proved in the assignment that if I have a cluster 00:21:05.460 --> 00:21:10.590 align:middle line:84% point of S, then there exists a sequence in S takeaway c 00:21:10.590 --> 00:21:11.790 align:middle line:90% that converges to c. 00:21:11.790 --> 00:21:14.690 align:middle line:90% 00:21:14.690 --> 00:21:24.560 align:middle line:84% Now, since-- say it this way-- by the previous theorem, 00:21:24.560 --> 00:21:27.950 align:middle line:84% this limit equals L1, this limit equals L2 if and only 00:21:27.950 --> 00:21:33.260 align:middle line:84% if for every sequence converging to c, f of x n converges to L1, 00:21:33.260 --> 00:21:34.955 align:middle line:90% g of x n converges to L2. 00:21:34.955 --> 00:21:41.640 align:middle line:90% 00:21:41.640 --> 00:21:43.860 align:middle line:84% By the previous theorem, we then conclude 00:21:43.860 --> 00:21:50.480 align:middle line:84% that L1 is equal to the limit as n goes to infinity-- 00:21:50.480 --> 00:21:53.790 align:middle line:84% actually getting ahead of myself a little bit. 00:21:53.790 --> 00:21:56.580 align:middle line:84% So let's pause right there and reset. 00:21:56.580 --> 00:21:58.730 align:middle line:84% So now we have the sequence converging to c. 00:21:58.730 --> 00:22:02.540 align:middle line:90% 00:22:02.540 --> 00:22:09.380 align:middle line:84% Then by the assumption here for all n, f of x n 00:22:09.380 --> 00:22:11.660 align:middle line:84% is less than or equal to g of x n. 00:22:11.660 --> 00:22:14.830 align:middle line:90% 00:22:14.830 --> 00:22:22.140 align:middle line:84% And since f of x n converges to L1, 00:22:22.140 --> 00:22:31.590 align:middle line:84% and g of x n converges to L2, we get by, again, 00:22:31.590 --> 00:22:34.530 align:middle line:84% this theorem about sequences, which says if I have two 00:22:34.530 --> 00:22:39.510 align:middle line:84% sequences, f of x sub n-- this is one sequence is less than 00:22:39.510 --> 00:22:43.080 align:middle line:84% or equal to another sequence, g of x n-- 00:22:43.080 --> 00:22:46.320 align:middle line:84% and they both converge, then the limits 00:22:46.320 --> 00:23:06.160 align:middle line:84% satisfy the same inequality, which 00:23:06.160 --> 00:23:07.300 align:middle line:90% is what we wanted to prove. 00:23:07.300 --> 00:23:27.820 align:middle line:90% 00:23:27.820 --> 00:23:31.900 align:middle line:84% So like I said, this theorem here 00:23:31.900 --> 00:23:35.510 align:middle line:90% follows from the analogous-- 00:23:35.510 --> 00:23:39.170 align:middle line:84% in fact, I should have written this out or in fact, 00:23:39.170 --> 00:23:40.240 align:middle line:90% I will write it out now. 00:23:40.240 --> 00:23:51.035 align:middle line:84% So the analogous statement for sequences was if for all n, 00:23:51.035 --> 00:23:57.690 align:middle line:84% a n is less than or equal to b n, then limit 00:23:57.690 --> 00:24:05.730 align:middle line:84% as n goes to infinity, assuming both limits exist, satisfy 00:24:05.730 --> 00:24:07.620 align:middle line:90% the same inequality. 00:24:07.620 --> 00:24:11.520 align:middle line:84% So this is an analogous theorem to this theorem, 00:24:11.520 --> 00:24:12.660 align:middle line:90% which we had for sequences. 00:24:12.660 --> 00:24:16.710 align:middle line:84% And we used this theorem from sequences to prove it. 00:24:16.710 --> 00:24:20.700 align:middle line:84% Now, following that same philosophy, 00:24:20.700 --> 00:24:22.170 align:middle line:84% you can prove analogous statements 00:24:22.170 --> 00:24:25.170 align:middle line:84% for functions, limits of functions, 00:24:25.170 --> 00:24:27.360 align:middle line:90% as you did from sequences. 00:24:27.360 --> 00:24:28.560 align:middle line:90% You get these for free. 00:24:28.560 --> 00:24:31.110 align:middle line:84% And instead of stating all of the theorems we 00:24:31.110 --> 00:24:34.680 align:middle line:84% did for sequences except now for limits of functions, 00:24:34.680 --> 00:24:39.940 align:middle line:84% I'm just going to quickly say you get the same thing. 00:24:39.940 --> 00:24:53.970 align:middle line:84% So by using the previous theorem, which 00:24:53.970 --> 00:24:59.850 align:middle line:84% connects convergence of functions 00:24:59.850 --> 00:25:21.850 align:middle line:84% to convergence of sequences, we have analogous theorems for-- 00:25:21.850 --> 00:25:23.485 align:middle line:90% and let me state it this way-- 00:25:23.485 --> 00:25:35.250 align:middle line:90% 00:25:35.250 --> 00:25:38.490 align:middle line:90% for limits of functions now. 00:25:38.490 --> 00:25:48.260 align:middle line:84% And for example, you have the squeeze theorem, 00:25:48.260 --> 00:25:49.490 align:middle line:90% namely if I have-- 00:25:49.490 --> 00:25:52.700 align:middle line:84% so just talking this out-- if I have three functions, 00:25:52.700 --> 00:25:55.250 align:middle line:84% say f is less than or equal to g is less than 00:25:55.250 --> 00:25:59.360 align:middle line:84% or equal to h, and f of x and this converges to L, 00:25:59.360 --> 00:26:03.680 align:middle line:84% h of x converges to L, then g of x converges to L. 00:26:03.680 --> 00:26:05.810 align:middle line:84% That's what I mean by analogous statement. 00:26:05.810 --> 00:26:08.480 align:middle line:90% 00:26:08.480 --> 00:26:19.040 align:middle line:84% You also have theorems about algebraic operations 00:26:19.040 --> 00:26:32.250 align:middle line:84% and limits, meaning that if I have two functions that 00:26:32.250 --> 00:26:35.520 align:middle line:84% have limits as x goes to c, then f plus g will 00:26:35.520 --> 00:26:37.020 align:middle line:90% have a limit as x goes to c. 00:26:37.020 --> 00:26:40.050 align:middle line:84% And the limit of the sum is the sum of the limits. 00:26:40.050 --> 00:26:46.587 align:middle line:84% Same thing with the product, and same thing with the quotient, 00:26:46.587 --> 00:26:48.420 align:middle line:84% assuming the limit on the bottom is nonzero. 00:26:48.420 --> 00:26:58.010 align:middle line:90% 00:26:58.010 --> 00:27:00.980 align:middle line:84% And then similarly, we also have a theorem 00:27:00.980 --> 00:27:06.680 align:middle line:84% about the absolute value and limits, namely if f of x 00:27:06.680 --> 00:27:11.780 align:middle line:84% converges to L as x goes to c, then the absolute value of f 00:27:11.780 --> 00:27:15.920 align:middle line:84% of x converges to the absolute value of L as x goes to c. 00:27:15.920 --> 00:27:19.490 align:middle line:84% So you have all of these analogous statements, 00:27:19.490 --> 00:27:23.270 align:middle line:84% or theorems that are analogous to the statements from what 00:27:23.270 --> 00:27:26.390 align:middle line:84% we did for sequences, but now for limits of functions 00:27:26.390 --> 00:27:28.580 align:middle line:90% as x goes to c. 00:27:28.580 --> 00:27:31.730 align:middle line:84% And I'm not going to state them all. 00:27:31.730 --> 00:27:33.920 align:middle line:84% You can see this in the textbook. 00:27:33.920 --> 00:27:37.360 align:middle line:84% Maybe the proof of some of them I'll give as exercises. 00:27:37.360 --> 00:27:41.630 align:middle line:90% 00:27:41.630 --> 00:27:45.980 align:middle line:84% Now let's separate the notion of a limit 00:27:45.980 --> 00:27:50.400 align:middle line:84% of a function from that of a sequence just a little bit. 00:27:50.400 --> 00:27:57.550 align:middle line:84% So unlike when we talk about limits of sequences, 00:27:57.550 --> 00:28:02.080 align:middle line:84% here we're letting a point x converge or get close 00:28:02.080 --> 00:28:04.460 align:middle line:84% to a point c, but there's two ways 00:28:04.460 --> 00:28:07.750 align:middle line:84% it can get close to c on the real number line. 00:28:07.750 --> 00:28:10.570 align:middle line:84% It can converge to c from the left 00:28:10.570 --> 00:28:13.660 align:middle line:84% or it can converge to c from the right. 00:28:13.660 --> 00:28:16.302 align:middle line:84% And this leads to the notion of left and right limits 00:28:16.302 --> 00:28:16.885 align:middle line:90% of a function. 00:28:16.885 --> 00:28:35.450 align:middle line:90% 00:28:35.450 --> 00:28:41.510 align:middle line:84% So start the definition here and we'll go to the next board-- 00:28:41.510 --> 00:28:59.750 align:middle line:84% so let S be a subset of R, and suppose c 00:28:59.750 --> 00:29:10.590 align:middle line:84% is a cluster point of minus infinity, 00:29:10.590 --> 00:29:17.380 align:middle line:90% 0 intersect of c intersect S. 00:29:17.380 --> 00:29:19.450 align:middle line:84% So what I'm defining now is the notion 00:29:19.450 --> 00:29:21.910 align:middle line:84% of a function converging to something 00:29:21.910 --> 00:29:24.200 align:middle line:90% as x goes to c from the left. 00:29:24.200 --> 00:29:28.030 align:middle line:84% That's why I'm looking at only S intersect minus infinity to c. 00:29:28.030 --> 00:29:29.770 align:middle line:84% So I'm only looking to the left of c. 00:29:29.770 --> 00:29:33.540 align:middle line:90% 00:29:33.540 --> 00:29:41.220 align:middle line:84% We say f of x converges to L as x converges to c from the left 00:29:41.220 --> 00:29:54.610 align:middle line:84% by putting a minus sign up here if a similar definition as that 00:29:54.610 --> 00:29:59.410 align:middle line:84% of the limit, but now we only look 00:29:59.410 --> 00:30:01.360 align:middle line:84% at x getting close to c from the left. 00:30:01.360 --> 00:30:02.890 align:middle line:84% If for all epsilon positive there 00:30:02.890 --> 00:30:11.470 align:middle line:84% exists a delta positive such that for all 00:30:11.470 --> 00:30:21.660 align:middle line:84% x in S satisfying c minus delta is less than x is less than c-- 00:30:21.660 --> 00:30:27.200 align:middle line:84% so it's close to c but to the left of c-- 00:30:27.200 --> 00:30:33.140 align:middle line:84% we have that f of x minus L is less than epsilon. 00:30:33.140 --> 00:30:40.730 align:middle line:84% And in this case, we write the limit 00:30:40.730 --> 00:30:57.160 align:middle line:84% as x goes to c with the minus sign up top, f of x equals L. 00:30:57.160 --> 00:30:59.320 align:middle line:84% And then we have an analogous definition 00:30:59.320 --> 00:31:02.530 align:middle line:84% for converging to or taking a limit of a function 00:31:02.530 --> 00:31:04.030 align:middle line:90% as x goes to c from the right. 00:31:04.030 --> 00:31:08.570 align:middle line:90% 00:31:08.570 --> 00:31:19.740 align:middle line:84% If c is a cluster point of c infinity, 00:31:19.740 --> 00:31:24.037 align:middle line:90% c comma infinity intersect S-- 00:31:24.037 --> 00:31:24.995 align:middle line:90% so I shouldn't say if-- 00:31:24.995 --> 00:31:27.680 align:middle line:90% 00:31:27.680 --> 00:31:32.285 align:middle line:84% [INAUDIBLE] is a cluster point of, now just taking S 00:31:32.285 --> 00:31:43.140 align:middle line:84% that's to the right of c, then we say that f of x converges 00:31:43.140 --> 00:31:50.760 align:middle line:84% to L as x converges to c plus, meaning as x converges to c 00:31:50.760 --> 00:31:56.550 align:middle line:84% from the right, if for all epsilon positive there exists 00:31:56.550 --> 00:32:04.740 align:middle line:84% a delta positive such that for all x in S satisfying c is less 00:32:04.740 --> 00:32:07.170 align:middle line:84% than x is less than c plus delta-- 00:32:07.170 --> 00:32:10.890 align:middle line:84% so now it's close to c but to the right of c-- 00:32:10.890 --> 00:32:13.670 align:middle line:90% 00:32:13.670 --> 00:32:19.420 align:middle line:84% we have that f of x minus L is less than epsilon. 00:32:19.420 --> 00:32:24.340 align:middle line:84% And similarly to the notation up there, 00:32:24.340 --> 00:32:38.490 align:middle line:84% we write limit as x goes to c plus of f of x equals L. 00:32:38.490 --> 00:32:44.040 align:middle line:84% Now, just like we proved this theorem for limits, 00:32:44.040 --> 00:32:48.150 align:middle line:84% you can state and prove an analogous statement 00:32:48.150 --> 00:32:50.110 align:middle line:90% for one-sided limits. 00:32:50.110 --> 00:32:55.960 align:middle line:84% So such a statement would be, for example, 00:32:55.960 --> 00:32:58.860 align:middle line:84% limit as x goes to c minus of f of x equals L 00:32:58.860 --> 00:33:01.050 align:middle line:84% if and only if for every sequence 00:33:01.050 --> 00:33:06.930 align:middle line:84% x sub n satisfying x sub n is less than c converging to c, 00:33:06.930 --> 00:33:13.700 align:middle line:84% we have f of x n converges to L. So these two just kind of limit 00:33:13.700 --> 00:33:17.120 align:middle line:84% how f behaves near a point c if we're just 00:33:17.120 --> 00:33:22.970 align:middle line:84% looking to the left of c or to the right of c, but not at c. 00:33:22.970 --> 00:33:26.870 align:middle line:90% So let's, for example, look at-- 00:33:26.870 --> 00:33:29.975 align:middle line:90% 00:33:29.975 --> 00:33:32.600 align:middle line:84% I think this is usually referred to as the heavy side function. 00:33:32.600 --> 00:33:36.760 align:middle line:84% So f of x equals 0 for x less than or equal to 0, 00:33:36.760 --> 00:33:43.610 align:middle line:84% and 1 if x is bigger than or equal to 0. 00:33:43.610 --> 00:33:52.760 align:middle line:90% So the graph is like that. 00:33:52.760 --> 00:33:55.380 align:middle line:90% 00:33:55.380 --> 00:33:58.530 align:middle line:84% And why do people care about this function? 00:33:58.530 --> 00:34:01.740 align:middle line:84% Well, in a certain sense, if you take the derivative 00:34:01.740 --> 00:34:04.830 align:middle line:84% of this function, you get what's called the Dirac delta 00:34:04.830 --> 00:34:06.550 align:middle line:84% function, although that's not a function. 00:34:06.550 --> 00:34:07.620 align:middle line:90% That's a distribution. 00:34:07.620 --> 00:34:10.429 align:middle line:90% 00:34:10.429 --> 00:34:13.190 align:middle line:84% But that's why this function has a name attached 00:34:13.190 --> 00:34:15.199 align:middle line:84% to it because if you take its derivative, 00:34:15.199 --> 00:34:21.210 align:middle line:84% you get something somewhat special. 00:34:21.210 --> 00:34:23.199 align:middle line:84% Anyways, we're not even at derivatives. 00:34:23.199 --> 00:34:25.199 align:middle line:84% We're not even going to talk about distributions 00:34:25.199 --> 00:34:25.870 align:middle line:90% in this class. 00:34:25.870 --> 00:34:29.199 align:middle line:84% So let's get back to one-sided limits. 00:34:29.199 --> 00:34:37.290 align:middle line:84% So if I look at this function for x close to 0 from the left, 00:34:37.290 --> 00:34:38.140 align:middle line:90% f is just 0. 00:34:38.140 --> 00:34:46.469 align:middle line:90% 00:34:46.469 --> 00:34:51.510 align:middle line:84% So in fact, since I'm only looking at x to the left of 0 00:34:51.510 --> 00:34:57.810 align:middle line:84% in this limit, this is just plugging in x less than 0, 00:34:57.810 --> 00:34:59.400 align:middle line:90% so I get 0. 00:34:59.400 --> 00:35:00.150 align:middle line:90% And that's just 0. 00:35:00.150 --> 00:35:03.000 align:middle line:90% 00:35:03.000 --> 00:35:11.860 align:middle line:84% And if I look at this function from the right, 00:35:11.860 --> 00:35:15.010 align:middle line:84% to the right of 0, then f of x is just 1. 00:35:15.010 --> 00:35:16.060 align:middle line:90% It's just 1 identically. 00:35:16.060 --> 00:35:16.930 align:middle line:90% And so I get-- 00:35:16.930 --> 00:35:21.700 align:middle line:90% 00:35:21.700 --> 00:35:23.680 align:middle line:84% and it's, again, although I haven't 00:35:23.680 --> 00:35:28.810 align:middle line:84% shown that one-sided limits of constants equal to constant, 00:35:28.810 --> 00:35:30.790 align:middle line:84% I think that should be something you can easily 00:35:30.790 --> 00:35:33.380 align:middle line:90% believe or write out yourself. 00:35:33.380 --> 00:35:36.700 align:middle line:84% So for this function, we see that it 00:35:36.700 --> 00:35:39.910 align:middle line:84% does have two-sided limits, except those limits don't 00:35:39.910 --> 00:35:42.010 align:middle line:90% equal each other. 00:35:42.010 --> 00:35:47.980 align:middle line:84% And they certainly don't equal-- so this one does equal f of 0, 00:35:47.980 --> 00:35:52.697 align:middle line:84% but I could have made f of 0 to be a 1/2, 00:35:52.697 --> 00:35:54.280 align:middle line:84% and then this still would have been 0, 00:35:54.280 --> 00:35:57.130 align:middle line:84% and not equal to the function evaluated at the point. 00:35:57.130 --> 00:36:04.930 align:middle line:84% So again, I'm making this point that for limits, just limits, 00:36:04.930 --> 00:36:09.210 align:middle line:84% it does not matter what the function is doing at the point. 00:36:09.210 --> 00:36:11.400 align:middle line:84% Limit only cares about how a function 00:36:11.400 --> 00:36:12.705 align:middle line:90% behaves near a a point. 00:36:12.705 --> 00:36:16.440 align:middle line:84% One-sided limits augment that by saying 00:36:16.440 --> 00:36:19.830 align:middle line:84% we're only going to care about the function near the point 00:36:19.830 --> 00:36:22.980 align:middle line:84% and to the left for the left limit, 00:36:22.980 --> 00:36:25.740 align:middle line:84% and to the right for the right limit. 00:36:25.740 --> 00:36:28.500 align:middle line:84% So I didn't say that, but this we call the left limit, 00:36:28.500 --> 00:36:30.330 align:middle line:84% this we call the right limit, simply 00:36:30.330 --> 00:36:33.300 align:middle line:84% because we're getting close to c from the right 00:36:33.300 --> 00:36:34.290 align:middle line:90% and from the left. 00:36:34.290 --> 00:36:45.810 align:middle line:90% 00:36:45.810 --> 00:36:53.720 align:middle line:84% But what is the connection between left and right limits? 00:36:53.720 --> 00:36:55.030 align:middle line:90% We have the following-- 00:36:55.030 --> 00:37:01.000 align:middle line:90% 00:37:01.000 --> 00:37:07.720 align:middle line:84% let S be a subset of R, f be a function from S to R, 00:37:07.720 --> 00:37:14.410 align:middle line:84% and suppose that c is a cluster point 00:37:14.410 --> 00:37:20.930 align:middle line:84% of both sets minus infinity to c intersect S. 00:37:20.930 --> 00:37:27.110 align:middle line:84% And c infinity intersect S. That way 00:37:27.110 --> 00:37:32.870 align:middle line:84% I can talk about the left and right limits of c or at c. 00:37:32.870 --> 00:37:37.640 align:middle line:90% 00:37:37.640 --> 00:37:44.340 align:middle line:84% Then-- so first off, if c is a cluster 00:37:44.340 --> 00:37:46.200 align:middle line:84% point of any one of these sets, it's 00:37:46.200 --> 00:37:48.600 align:middle line:84% going to be a cluster point of the set S. 00:37:48.600 --> 00:37:51.940 align:middle line:84% So we can actually look at the limit. 00:37:51.940 --> 00:37:54.150 align:middle line:84% So then the limit as x goes to c of f 00:37:54.150 --> 00:37:57.720 align:middle line:84% of x exists and equals L if and only 00:37:57.720 --> 00:38:03.900 align:middle line:84% if the limit as x goes to c from the left of f of x 00:38:03.900 --> 00:38:10.110 align:middle line:84% equals the limit as x goes to c from the right of f of x equals 00:38:10.110 --> 00:38:18.250 align:middle line:90% L. 00:38:18.250 --> 00:38:20.240 align:middle line:84% So this kind of looks like the theorem 00:38:20.240 --> 00:38:22.090 align:middle line:84% we proved about lim sup and lim inf, 00:38:22.090 --> 00:38:25.930 align:middle line:84% but they don't have any connection. 00:38:25.930 --> 00:38:35.800 align:middle line:90% 00:38:35.800 --> 00:38:39.100 align:middle line:84% If you want to make some connection between the way 00:38:39.100 --> 00:38:41.320 align:middle line:84% this theorem looks and the statement of the theorem 00:38:41.320 --> 00:38:46.180 align:middle line:84% for lim inf and lim sup, so this is kind of saying that limit 00:38:46.180 --> 00:38:49.240 align:middle line:84% of a function equals L if we approach from the left 00:38:49.240 --> 00:38:53.800 align:middle line:84% or right the function f approaches L. 00:38:53.800 --> 00:38:57.850 align:middle line:84% And for the lim sup lim inf guy that we did for sequences, 00:38:57.850 --> 00:39:00.430 align:middle line:84% you could take that as saying that the limit of a sequence 00:39:00.430 --> 00:39:04.630 align:middle line:84% equals L if and only if following 00:39:04.630 --> 00:39:08.440 align:middle line:84% the sequence from below, that approaches L, 00:39:08.440 --> 00:39:12.310 align:middle line:84% and following the sequence from above, that also approaches L. 00:39:12.310 --> 00:39:14.448 align:middle line:84% So there's kind of two directions 00:39:14.448 --> 00:39:16.240 align:middle line:84% there, just as there's two directions here, 00:39:16.240 --> 00:39:17.680 align:middle line:90% but not really. 00:39:17.680 --> 00:39:24.580 align:middle line:90% 00:39:24.580 --> 00:39:31.350 align:middle line:84% So let's give a quick proof of this theorem. 00:39:31.350 --> 00:39:36.130 align:middle line:90% 00:39:36.130 --> 00:39:38.530 align:middle line:84% It's not difficult. It follows almost immediately 00:39:38.530 --> 00:39:44.850 align:middle line:90% from the definitions. 00:39:44.850 --> 00:39:54.760 align:middle line:90% 00:39:54.760 --> 00:39:58.820 align:middle line:84% In fact, I'm going to do only one direction. 00:39:58.820 --> 00:40:04.420 align:middle line:84% So this direction should, assuming this, and proving 00:40:04.420 --> 00:40:07.810 align:middle line:90% this should be pretty clear. 00:40:07.810 --> 00:40:13.385 align:middle line:84% If I have this, this means that if I want to be close to L, 00:40:13.385 --> 00:40:15.435 align:middle line:90% I just need to be close to c. 00:40:15.435 --> 00:40:17.060 align:middle line:84% And therefore, it doesn't matter if I'm 00:40:17.060 --> 00:40:18.530 align:middle line:84% close to c from the left or right. 00:40:18.530 --> 00:40:21.600 align:middle line:90% I'll be close to L. 00:40:21.600 --> 00:40:24.680 align:middle line:84% And going this direction, this says 00:40:24.680 --> 00:40:26.750 align:middle line:84% I just need to be close to c from the left, 00:40:26.750 --> 00:40:28.520 align:middle line:84% sufficiently close, and I need to be 00:40:28.520 --> 00:40:30.920 align:middle line:84% close to c from the right, sufficiently close, 00:40:30.920 --> 00:40:37.510 align:middle line:84% to be close to L. So really, there's 00:40:37.510 --> 00:40:42.170 align:middle line:84% not a lot of trickiness in the proof. 00:40:42.170 --> 00:40:43.810 align:middle line:84% It's just writing these things out. 00:40:43.810 --> 00:40:46.150 align:middle line:84% And so I'm just going to write out one direction 00:40:46.150 --> 00:40:48.850 align:middle line:84% and leave the other direction to you. 00:40:48.850 --> 00:41:00.630 align:middle line:84% So let's assume that the left limit equals 00:41:00.630 --> 00:41:07.420 align:middle line:84% the right limit equals L. And now we 00:41:07.420 --> 00:41:09.640 align:middle line:84% want to show that the limit as x goes 00:41:09.640 --> 00:41:21.440 align:middle line:84% to c of f of x equals L. Now we want to show limit 00:41:21.440 --> 00:41:24.890 align:middle line:84% as x goes to c of f of x equals L. 00:41:24.890 --> 00:41:26.990 align:middle line:84% So let's go back to the definition. 00:41:26.990 --> 00:41:28.190 align:middle line:90% Let epsilon be positive. 00:41:28.190 --> 00:41:32.510 align:middle line:84% We want to be able to find delta so that f of x 00:41:32.510 --> 00:41:37.085 align:middle line:84% is within epsilon to L if x is within delta to c. 00:41:37.085 --> 00:41:39.740 align:middle line:90% 00:41:39.740 --> 00:41:41.090 align:middle line:90% And what's the point? 00:41:41.090 --> 00:41:50.950 align:middle line:84% Here's c and here's L. So this is the picture 00:41:50.950 --> 00:41:53.370 align:middle line:90% that goes along with this. 00:41:53.370 --> 00:41:55.290 align:middle line:84% Assuming these two limits equal L, 00:41:55.290 --> 00:41:59.370 align:middle line:84% I know that there exists a delta 1 so 00:41:59.370 --> 00:42:06.732 align:middle line:84% that if I'm in this interval, then here's 00:42:06.732 --> 00:42:12.300 align:middle line:90% L plus epsilon, L minus epsilon. 00:42:12.300 --> 00:42:16.980 align:middle line:84% Then if I'm within delta 1 to c and to the left, 00:42:16.980 --> 00:42:33.200 align:middle line:84% then f will be close to L in that interval. 00:42:33.200 --> 00:42:36.830 align:middle line:84% And then since the limit as x goes to c from the right equals 00:42:36.830 --> 00:42:40.460 align:middle line:84% L, there exists some delta 2 so that if I'm in this interval 00:42:40.460 --> 00:42:46.940 align:middle line:90% then I'll be, again, close to L. 00:42:46.940 --> 00:42:52.115 align:middle line:84% But this means that if I choose the smaller of these two, 00:42:52.115 --> 00:42:53.490 align:middle line:84% and I look at the whole interval, 00:42:53.490 --> 00:42:56.260 align:middle line:84% then f will be close to L on the whole interval. 00:42:56.260 --> 00:42:57.840 align:middle line:90% And that's it. 00:42:57.840 --> 00:42:59.550 align:middle line:90% So let epsilon be positive. 00:42:59.550 --> 00:43:02.430 align:middle line:90% 00:43:02.430 --> 00:43:09.290 align:middle line:84% Then limit as x goes to c minus f of x equals 00:43:09.290 --> 00:43:14.420 align:middle line:84% L, this implies there exists delta 1 positive such 00:43:14.420 --> 00:43:24.520 align:middle line:84% that if x minus c is less than delta 1, then 00:43:24.520 --> 00:43:33.620 align:middle line:84% I get that f of x c minus delta 1 is less than x is less delta 00:43:33.620 --> 00:43:35.510 align:middle line:90% is less than c. 00:43:35.510 --> 00:43:40.020 align:middle line:84% This implies that f of x minus L is less than epsilon. 00:43:40.020 --> 00:43:43.730 align:middle line:84% And similarly for the right limit, 00:43:43.730 --> 00:43:51.340 align:middle line:84% since limit as x goes to c plus of f of x equals L, 00:43:51.340 --> 00:43:59.630 align:middle line:84% this implies, by the definition, there exists a delta 2 positive 00:43:59.630 --> 00:44:08.270 align:middle line:84% such that if c is less than x is less than c plus delta 2, 00:44:08.270 --> 00:44:15.960 align:middle line:84% then I get f of x minus L is less than epsilon. 00:44:15.960 --> 00:44:18.670 align:middle line:90% 00:44:18.670 --> 00:44:29.330 align:middle line:84% Now choose delta to be the minimum of delta 1 and delta 2. 00:44:29.330 --> 00:44:31.295 align:middle line:84% And then we'll now show that this delta works. 00:44:31.295 --> 00:44:42.600 align:middle line:90% 00:44:42.600 --> 00:44:44.655 align:middle line:84% So now we're going to show this delta works. 00:44:44.655 --> 00:44:49.820 align:middle line:90% 00:44:49.820 --> 00:45:05.240 align:middle line:84% So if less than delta, then this implies that either x 00:45:05.240 --> 00:45:07.745 align:middle line:90% is in c minus delta-- 00:45:07.745 --> 00:45:13.880 align:middle line:90% 00:45:13.880 --> 00:45:15.640 align:middle line:84% so if we take something close within delta 00:45:15.640 --> 00:45:20.800 align:middle line:84% to c, and delta is the minimum of these two distances, then-- 00:45:20.800 --> 00:45:22.780 align:middle line:84% for the sake of this picture, let's say 00:45:22.780 --> 00:45:26.110 align:middle line:84% delta 1 equals delta 2, so now I'm looking at this interval-- 00:45:26.110 --> 00:45:29.260 align:middle line:90% then two cases. 00:45:29.260 --> 00:45:32.800 align:middle line:84% Either x is in c minus delta c, which 00:45:32.800 --> 00:45:37.360 align:middle line:84% is a subset of c minus delta 1 c since delta 00:45:37.360 --> 00:45:44.800 align:middle line:84% is the minimum of those two deltas, which implies 00:45:44.800 --> 00:45:57.330 align:middle line:84% by the first inequality here for delta 1 or x is in c, c 00:45:57.330 --> 00:46:00.600 align:middle line:84% plus delta, which is a subset of c, c 00:46:00.600 --> 00:46:09.340 align:middle line:84% plus delta 2, which by our choice of delta 2 00:46:09.340 --> 00:46:10.090 align:middle line:90% gives us that. 00:46:10.090 --> 00:46:14.280 align:middle line:90% 00:46:14.280 --> 00:46:21.280 align:middle line:84% Thus we've shown that if x minus c is less than delta, 00:46:21.280 --> 00:46:26.880 align:middle line:84% then f of x minus L is less than epsilon. 00:46:26.880 --> 00:46:29.150 align:middle line:90% That's the end of the proof. 00:46:29.150 --> 00:46:32.840 align:middle line:84% So I've said this over and over again-- 00:46:32.840 --> 00:46:36.290 align:middle line:84% limits of functions don't care about what the function is 00:46:36.290 --> 00:46:37.235 align:middle line:90% doing at the point. 00:46:37.235 --> 00:46:40.580 align:middle line:84% It cares about what the function is doing near the point. 00:46:40.580 --> 00:46:51.450 align:middle line:84% Now we're going to discuss the notion of continuity, 00:46:51.450 --> 00:46:58.360 align:middle line:84% which connects the limit of a function at a point 00:46:58.360 --> 00:47:00.140 align:middle line:90% to the function. 00:47:00.140 --> 00:47:02.620 align:middle line:84% So it connects how a function behaves 00:47:02.620 --> 00:47:07.480 align:middle line:84% near a point to the function evaluated at the point. 00:47:07.480 --> 00:47:18.520 align:middle line:84% And so you can even write this down. 00:47:18.520 --> 00:47:25.150 align:middle line:90% 00:47:25.150 --> 00:47:33.670 align:middle line:84% How a function behaves near a point compared to-- 00:47:33.670 --> 00:47:40.660 align:middle line:90% 00:47:40.660 --> 00:47:43.285 align:middle line:90% so near, at. 00:47:43.285 --> 00:47:50.800 align:middle line:90% 00:47:50.800 --> 00:47:56.560 align:middle line:84% And you'll see when I write down the definition, basically, 00:47:56.560 --> 00:47:58.390 align:middle line:84% what's staring you in the face is 00:47:58.390 --> 00:48:00.760 align:middle line:84% that the definition of continuity 00:48:00.760 --> 00:48:05.710 align:middle line:84% is that the limit as x goes to c of f of x equals f of c. 00:48:05.710 --> 00:48:07.495 align:middle line:84% So we had these examples where the limit-- 00:48:07.495 --> 00:48:10.170 align:middle line:90% 00:48:10.170 --> 00:48:13.750 align:middle line:84% I think already erased it, but where the limit exists 00:48:13.750 --> 00:48:17.090 align:middle line:90% but does not equal f of c. 00:48:17.090 --> 00:48:21.380 align:middle line:84% And here for continuity, the notion of continuity 00:48:21.380 --> 00:48:24.560 align:middle line:84% is that the limit as x goes to c of f of x actually 00:48:24.560 --> 00:48:27.210 align:middle line:84% equals the function evaluated at that point. 00:48:27.210 --> 00:48:33.390 align:middle line:84% So we have the following definition. 00:48:33.390 --> 00:48:42.120 align:middle line:84% Let S be a subset of R and c an element of S. 00:48:42.120 --> 00:48:56.860 align:middle line:84% We say f is continuous at c if for all epsilon positive, 00:48:56.860 --> 00:48:58.840 align:middle line:84% there exists a delta positive such 00:48:58.840 --> 00:49:08.480 align:middle line:84% that for all x and S satisfying x minus c is less than delta-- 00:49:08.480 --> 00:49:12.740 align:middle line:84% so in particular, I can now, for example, 00:49:12.740 --> 00:49:15.230 align:middle line:84% x equals c will satisfy this inequality. 00:49:15.230 --> 00:49:17.110 align:middle line:84% I don't have the 0 is less than that. 00:49:17.110 --> 00:49:21.060 align:middle line:90% 00:49:21.060 --> 00:49:24.330 align:middle line:84% So for all x and S which are close to c within delta, 00:49:24.330 --> 00:49:27.435 align:middle line:90% I have that f of x will be-- 00:49:27.435 --> 00:49:30.950 align:middle line:90% 00:49:30.950 --> 00:49:33.070 align:middle line:90% with an epsilon of f of c. 00:49:33.070 --> 00:49:38.950 align:middle line:90% 00:49:38.950 --> 00:49:46.580 align:middle line:84% So in this case, just a little f is 00:49:46.580 --> 00:49:54.240 align:middle line:84% continuous at every point on its domain that we're considering. 00:49:54.240 --> 00:49:55.860 align:middle line:90% We just say f is continuous. 00:49:55.860 --> 00:50:13.870 align:middle line:90% 00:50:13.870 --> 00:50:28.450 align:middle line:84% So for a function to be continuous at a point nearby 00:50:28.450 --> 00:50:33.190 align:middle line:84% x, x being near c, should mean that f of x 00:50:33.190 --> 00:50:34.900 align:middle line:90% should be near f of c. 00:50:34.900 --> 00:50:42.840 align:middle line:90% 00:50:42.840 --> 00:50:44.863 align:middle line:84% And let's go through some examples. 00:50:44.863 --> 00:50:46.530 align:middle line:84% Remember, whenever you get a definition, 00:50:46.530 --> 00:50:51.810 align:middle line:84% you should look for examples and then potentially negate it. 00:50:51.810 --> 00:50:56.800 align:middle line:84% We'll negate this definition in just a second 00:50:56.800 --> 00:50:59.860 align:middle line:84% to show that a function I wrote down a minute ago 00:50:59.860 --> 00:51:01.000 align:middle line:90% is not continuous. 00:51:01.000 --> 00:51:06.810 align:middle line:90% 00:51:06.810 --> 00:51:19.100 align:middle line:84% So the affine function f of x equals a times x plus b-- 00:51:19.100 --> 00:51:21.650 align:middle line:90% 00:51:21.650 --> 00:51:25.820 align:middle line:84% so S is R, so x is a real number-- 00:51:25.820 --> 00:51:29.780 align:middle line:84% is a continuous function, meaning 00:51:29.780 --> 00:51:33.260 align:middle line:84% it's continuous at every real number c. 00:51:33.260 --> 00:51:34.265 align:middle line:90% So let's prove this. 00:51:34.265 --> 00:51:38.830 align:middle line:90% 00:51:38.830 --> 00:51:40.720 align:middle line:90% Let c be an element of R. 00:51:40.720 --> 00:51:42.880 align:middle line:84% We want to show f is continuous at c. 00:51:42.880 --> 00:51:46.874 align:middle line:84% So we have to go through the definition. 00:51:46.874 --> 00:51:50.750 align:middle line:90% Let epsilon be positive. 00:51:50.750 --> 00:51:58.340 align:middle line:84% Choose delta to be epsilon over 1 plus the absolute value of a. 00:51:58.340 --> 00:52:01.250 align:middle line:84% And last time, in the previous lecture, 00:52:01.250 --> 00:52:06.110 align:middle line:84% I gave the intuition on why you would choose this delta based 00:52:06.110 --> 00:52:11.980 align:middle line:90% on the function and epsilon. 00:52:11.980 --> 00:52:13.113 align:middle line:90% I did a computation here. 00:52:13.113 --> 00:52:14.780 align:middle line:84% I'm just going to choose delta this way, 00:52:14.780 --> 00:52:16.010 align:middle line:90% and you'll see that it works. 00:52:16.010 --> 00:52:18.550 align:middle line:84% So now we have to show this delta works. 00:52:18.550 --> 00:52:23.410 align:middle line:84% If x minus c is less than delta, we 00:52:23.410 --> 00:52:27.590 align:middle line:84% should be able to now show that f of x minus f of c 00:52:27.590 --> 00:52:28.465 align:middle line:90% is less than epsilon. 00:52:28.465 --> 00:52:32.410 align:middle line:90% 00:52:32.410 --> 00:52:35.850 align:middle line:84% This is equal to a x plus b minus a c plus b. 00:52:35.850 --> 00:52:39.692 align:middle line:84% So this is equal to a times x minus c, 00:52:39.692 --> 00:52:42.680 align:middle line:84% which equals absolute value of a times absolute value 00:52:42.680 --> 00:52:44.090 align:middle line:90% of x minus c. 00:52:44.090 --> 00:52:49.240 align:middle line:90% This is less than delta. 00:52:49.240 --> 00:52:52.870 align:middle line:84% Absolute value of x minus c is less than delta times a, 00:52:52.870 --> 00:52:56.990 align:middle line:84% which equals the absolute value of a over 1 00:52:56.990 --> 00:53:00.110 align:middle line:84% plus the absolute value of a times epsilon, 00:53:00.110 --> 00:53:03.350 align:middle line:90% which is less than epsilon. 00:53:03.350 --> 00:53:06.830 align:middle line:84% Because a number over 1 plus that number 00:53:06.830 --> 00:53:08.330 align:middle line:90% is always less than 1. 00:53:08.330 --> 00:53:11.457 align:middle line:90% 00:53:11.457 --> 00:53:13.040 align:middle line:84% Maybe you were wondering, why didn't I 00:53:13.040 --> 00:53:17.381 align:middle line:84% just choose delta to be epsilon over the absolute value of a? 00:53:17.381 --> 00:53:20.300 align:middle line:84% This is just a smidgen of sophistication, 00:53:20.300 --> 00:53:22.310 align:middle line:84% that what happens if the absolute value of a 00:53:22.310 --> 00:53:23.220 align:middle line:90% is equal to 0? 00:53:23.220 --> 00:53:24.950 align:middle line:90% Then we would have divided by 0. 00:53:24.950 --> 00:53:31.040 align:middle line:84% So adding a 1 there takes care of that. 00:53:31.040 --> 00:53:36.590 align:middle line:84% So this guy is continuous at every c. 00:53:36.590 --> 00:53:39.410 align:middle line:90% So this function is continuous. 00:53:39.410 --> 00:53:47.660 align:middle line:90% 00:53:47.660 --> 00:53:58.105 align:middle line:84% How about a function that's not continuous at a point? 00:53:58.105 --> 00:54:04.080 align:middle line:90% 00:54:04.080 --> 00:54:05.490 align:middle line:84% Make sure this is the next topic. 00:54:05.490 --> 00:54:20.040 align:middle line:90% 00:54:20.040 --> 00:54:21.615 align:middle line:90% Here is a non-example. 00:54:21.615 --> 00:54:24.270 align:middle line:90% 00:54:24.270 --> 00:54:30.230 align:middle line:84% The function f of x, which equals 1 if x equals 0, 00:54:30.230 --> 00:54:32.930 align:middle line:90% 2 if x is not equal to 0. 00:54:32.930 --> 00:54:43.462 align:middle line:90% 00:54:43.462 --> 00:54:44.295 align:middle line:90% That's the function. 00:54:44.295 --> 00:54:47.960 align:middle line:90% 00:54:47.960 --> 00:54:55.585 align:middle line:90% f is not continuous at 0. 00:54:55.585 --> 00:54:58.400 align:middle line:90% 00:54:58.400 --> 00:55:07.280 align:middle line:84% So to prove this, let's negate the definition of continuity. 00:55:07.280 --> 00:55:12.800 align:middle line:84% So the negation of continuity is-- 00:55:12.800 --> 00:55:21.290 align:middle line:90% so f is not continuous at c if-- 00:55:21.290 --> 00:55:25.960 align:middle line:84% so the "for alls" become "there exists," and "there exists" 00:55:25.960 --> 00:55:26.830 align:middle line:90% become "for all." 00:55:26.830 --> 00:55:30.840 align:middle line:84% So if there exists some bad epsilon 00:55:30.840 --> 00:55:43.440 align:middle line:84% so that for all delta positive, there exist an x such 00:55:43.440 --> 00:55:47.220 align:middle line:84% that x minus c is less than delta, 00:55:47.220 --> 00:55:59.757 align:middle line:84% and we do not have the second inequality, f of x minus f of c 00:55:59.757 --> 00:56:01.590 align:middle line:84% is bigger than or equal to this bad epsilon. 00:56:01.590 --> 00:56:08.030 align:middle line:90% 00:56:08.030 --> 00:56:11.870 align:middle line:84% Now for this guy, it's pretty clear which x to choose. 00:56:11.870 --> 00:56:14.960 align:middle line:84% So let's think this out for a minute. 00:56:14.960 --> 00:56:17.060 align:middle line:84% There should be some bad epsilon 0 so 00:56:17.060 --> 00:56:23.210 align:middle line:84% that if I take any small interval around 0, 00:56:23.210 --> 00:56:25.620 align:middle line:84% I can find a point in this interval 00:56:25.620 --> 00:56:31.220 align:middle line:84% so that f of x minus f of c is going 00:56:31.220 --> 00:56:33.860 align:middle line:84% to be bigger than or equal to epsilon 0. 00:56:33.860 --> 00:56:40.300 align:middle line:84% Now here, f of c is f of 0, which is 1. 00:56:40.300 --> 00:56:45.520 align:middle line:84% Now what would be the bad epsilon so that f of 1 00:56:45.520 --> 00:56:50.260 align:middle line:84% is greater than distance 1 or greater than distance epsilon 0 00:56:50.260 --> 00:56:54.710 align:middle line:84% to f of x for some x in this interval? 00:56:54.710 --> 00:56:59.620 align:middle line:84% Well, if I take any x in this interval other than 0, 00:56:59.620 --> 00:57:01.840 align:middle line:84% and stick it into f, I'm going to get 2. 00:57:01.840 --> 00:57:07.600 align:middle line:84% And that's greater than or equal to distance 1 to f of 0. 00:57:07.600 --> 00:57:10.810 align:middle line:84% So epsilon 0 I will choose to be 1. 00:57:10.810 --> 00:57:15.380 align:middle line:90% 00:57:15.380 --> 00:57:21.390 align:middle line:84% So now we want to prove that f, this function here, is not 00:57:21.390 --> 00:57:23.220 align:middle line:90% continuous at 0. 00:57:23.220 --> 00:57:26.370 align:middle line:84% So I'll tell you what the bad epsilon is. 00:57:26.370 --> 00:57:30.300 align:middle line:90% Choose epsilon 0 equals 1. 00:57:30.300 --> 00:57:34.620 align:middle line:84% So now we have to show this bad epsilon 0 is indeed bad. 00:57:34.620 --> 00:57:35.940 align:middle line:90% Let delta be positive. 00:57:35.940 --> 00:57:40.270 align:middle line:84% We have to now find a number in this interval 00:57:40.270 --> 00:57:46.230 align:middle line:84% so that f of x minus f of 0 is bigger than or equal to 1. 00:57:46.230 --> 00:57:49.980 align:middle line:84% And like I said a minute ago, if you take any x in this interval 00:57:49.980 --> 00:57:51.810 align:middle line:84% other than 0 and stick it into f, 00:57:51.810 --> 00:57:55.725 align:middle line:84% I get 2, which is distance 1 to f of 0. 00:57:55.725 --> 00:58:00.660 align:middle line:90% 00:58:00.660 --> 00:58:03.440 align:middle line:90% Let x to be delta over 2, say. 00:58:03.440 --> 00:58:06.050 align:middle line:90% 00:58:06.050 --> 00:58:12.120 align:middle line:84% Then x minus 0 is less than delta. 00:58:12.120 --> 00:58:16.050 align:middle line:84% It's actually equal to delta over 2. 00:58:16.050 --> 00:58:20.970 align:middle line:90% And f of x minus f of 0-- 00:58:20.970 --> 00:58:27.440 align:middle line:84% this is equal to 2 minus 1, which is bigger than or equal 00:58:27.440 --> 00:58:29.960 align:middle line:90% to epsilon 0. 00:58:29.960 --> 00:58:34.310 align:middle line:84% So this function is not continuous. 00:58:34.310 --> 00:58:42.160 align:middle line:84% And so next time-- and I'll just leave this question here, 00:58:42.160 --> 00:58:43.980 align:middle line:84% which we'll address in the next lecture. 00:58:43.980 --> 00:58:46.270 align:middle line:84% But it's a kind of simple question. 00:58:46.270 --> 00:58:50.280 align:middle line:84% So first off, if you look at this function, 00:58:50.280 --> 00:58:53.070 align:middle line:84% it shouldn't be too hard to convince yourself, 00:58:53.070 --> 00:59:00.240 align:middle line:84% so you're also told when you were a child that a function is 00:59:00.240 --> 00:59:01.920 align:middle line:84% continuous If you can draw the graph 00:59:01.920 --> 00:59:06.500 align:middle line:84% and not lift up the pencil, which I better not see 00:59:06.500 --> 00:59:09.500 align:middle line:84% on the exam if I ask you about continuity. 00:59:09.500 --> 00:59:12.290 align:middle line:84% But anyway, for the sake of this conversation, 00:59:12.290 --> 00:59:15.750 align:middle line:84% let's take that as the intuition. 00:59:15.750 --> 00:59:17.900 align:middle line:84% So you can convince yourself that this function is 00:59:17.900 --> 00:59:19.970 align:middle line:90% continuous over here, though. 00:59:19.970 --> 00:59:24.170 align:middle line:84% If I'm getting close to-- let's say this is minus 1, 00:59:24.170 --> 00:59:28.790 align:middle line:84% then the function is getting close to 2. 00:59:28.790 --> 00:59:32.638 align:middle line:84% And the value of the function at minus 1 is 2. 00:59:32.638 --> 00:59:35.180 align:middle line:84% So the function is getting close to the value of the function 00:59:35.180 --> 00:59:36.290 align:middle line:90% at minus 1. 00:59:36.290 --> 00:59:39.575 align:middle line:84% And the same for if I'm looking at c equals 1. 00:59:39.575 --> 00:59:42.390 align:middle line:90% 00:59:42.390 --> 00:59:44.140 align:middle line:84% So you should be able to convince yourself 00:59:44.140 --> 00:59:46.780 align:middle line:84% that this function is continuous at every point 00:59:46.780 --> 00:59:49.810 align:middle line:90% other than at the origin 0. 00:59:49.810 --> 00:59:57.250 align:middle line:84% So a natural question to ask is, let f be a function, let's say, 00:59:57.250 --> 01:00:00.135 align:middle line:84% defined on the whole real number line. 01:00:00.135 --> 01:00:03.980 align:middle line:90% 01:00:03.980 --> 01:00:10.810 align:middle line:84% Does there exist a point in R such 01:00:10.810 --> 01:00:20.590 align:middle line:84% that f is continuous at this point c? 01:00:20.590 --> 01:00:23.500 align:middle line:84% For this example over here, we were able to-- 01:00:23.500 --> 01:00:27.490 align:middle line:84% any point other than 0, the function's continuous there. 01:00:27.490 --> 01:00:29.800 align:middle line:84% So natural question is, let's say 01:00:29.800 --> 01:00:32.620 align:middle line:90% I take an arbitrary function. 01:00:32.620 --> 01:00:36.330 align:middle line:84% Does it have to have a point where it's continuous? 01:00:36.330 --> 01:00:38.370 align:middle line:84% And next time, we'll see that answer is no. 01:00:38.370 --> 01:00:43.760 align:middle line:84% We'll give an example that's I think 01:00:43.760 --> 01:00:45.920 align:middle line:84% due to Dirichlet because it's named after him, 01:00:45.920 --> 01:00:48.740 align:middle line:84% but naming doesn't necessarily mean anything in math. 01:00:48.740 --> 01:00:51.410 align:middle line:84% Green's theorem is named after Green but he didn't prove it, 01:00:51.410 --> 01:00:55.920 align:middle line:84% so maybe it was due to somebody else. 01:00:55.920 --> 01:01:00.560 align:middle line:84% And we'll use a similar characterization of continuity 01:01:00.560 --> 01:01:02.840 align:middle line:84% that's kind of analogous to this first statement we 01:01:02.840 --> 01:01:04.370 align:middle line:90% had for limits. 01:01:04.370 --> 01:01:06.640 align:middle line:90% And we'll do that next time. 01:01:06.640 --> 01:01:13.000 align:middle line:90%