WEBVTT 00:00:00.000 --> 00:00:12.107 align:middle line:90% 00:00:12.107 --> 00:00:12.940 align:middle line:90% CASEY RODRIGUEZ: OK. 00:00:12.940 --> 00:00:17.170 align:middle line:84% So let's continue our discussion of continuity, which we began 00:00:17.170 --> 00:00:18.985 align:middle line:90% last time, which is-- 00:00:18.985 --> 00:00:26.250 align:middle line:90% 00:00:26.250 --> 00:00:30.550 align:middle line:84% which I defined last time, and I wrote again here, 00:00:30.550 --> 00:00:33.090 align:middle line:84% which intuitively says that if you want to be-- 00:00:33.090 --> 00:00:40.960 align:middle line:84% that if x is sufficiently close to number c, 00:00:40.960 --> 00:00:45.640 align:middle line:84% then f of x will be very close to f of c. 00:00:45.640 --> 00:00:48.700 align:middle line:84% So it connects how the function behaves 00:00:48.700 --> 00:00:55.120 align:middle line:84% near a point to how a function behaves at the point. 00:00:55.120 --> 00:00:59.070 align:middle line:84% And last time, we gave an example 00:00:59.070 --> 00:01:00.660 align:middle line:84% of a function which is continuous 00:01:00.660 --> 00:01:07.190 align:middle line:84% everywhere, namely the function f of x equals ax plus b, 00:01:07.190 --> 00:01:09.530 align:middle line:84% and one that was not continuous at a point c. 00:01:09.530 --> 00:01:10.370 align:middle line:90% And we ended with-- 00:01:10.370 --> 00:01:12.920 align:middle line:90% 00:01:12.920 --> 00:01:17.330 align:middle line:84% let me just recall the question I asked last time. 00:01:17.330 --> 00:01:22.940 align:middle line:84% If f is a function-- let's say that its whole domain is r. 00:01:22.940 --> 00:01:26.800 align:middle line:90% 00:01:26.800 --> 00:01:31.660 align:middle line:84% Does there exist a point where it's continuous? 00:01:31.660 --> 00:01:48.130 align:middle line:90% 00:01:48.130 --> 00:01:50.673 align:middle line:84% Now, I could answer this question now just 00:01:50.673 --> 00:01:52.840 align:middle line:84% using the definition, but I'll answer it in a minute 00:01:52.840 --> 00:01:56.350 align:middle line:84% after we prove the following theorem, which 00:01:56.350 --> 00:02:00.520 align:middle line:84% is an analog of this theorem that we proved here for limits. 00:02:00.520 --> 00:02:06.980 align:middle line:84% We showed that if you have a subset 00:02:06.980 --> 00:02:10.610 align:middle line:84% s of r, a cluster point of s-- so this 00:02:10.610 --> 00:02:13.610 align:middle line:84% is where you take limits, and then the limit 00:02:13.610 --> 00:02:16.010 align:middle line:84% as x goes to c of f of x equals l 00:02:16.010 --> 00:02:19.940 align:middle line:84% if and only if, for every sequence converging to c, 00:02:19.940 --> 00:02:23.870 align:middle line:84% we have f of xn, which is a new sequence, converges to l. 00:02:23.870 --> 00:02:27.680 align:middle line:84% So this connects limits of functions 00:02:27.680 --> 00:02:33.233 align:middle line:84% to limits of sequences-- limits along sequences, if you like. 00:02:33.233 --> 00:02:35.150 align:middle line:84% We're going to prove an analog of this theorem 00:02:35.150 --> 00:02:38.910 align:middle line:84% now for the notion of continuity. 00:02:38.910 --> 00:02:42.705 align:middle line:90% But there's kind of-- 00:02:42.705 --> 00:02:45.847 align:middle line:84% and these two definitions certainly look-- 00:02:45.847 --> 00:02:47.430 align:middle line:84% if you look at the definition of limit 00:02:47.430 --> 00:02:49.620 align:middle line:84% and the definition of continuity, 00:02:49.620 --> 00:02:52.350 align:middle line:84% this one certainly looks like the definition of limit 00:02:52.350 --> 00:02:55.830 align:middle line:84% where now the limit has to be f of c, the function evaluated 00:02:55.830 --> 00:02:56.670 align:middle line:90% at that point. 00:02:56.670 --> 00:03:05.000 align:middle line:84% And it essentially is, but we do have a degenerate case where 00:03:05.000 --> 00:03:07.760 align:middle line:84% c here is not required to be a cluster point of s. 00:03:07.760 --> 00:03:10.970 align:middle line:90% It's just any point of s. 00:03:10.970 --> 00:03:14.660 align:middle line:84% And we'll have two cases, c is a cluster point of s 00:03:14.660 --> 00:03:16.580 align:middle line:84% and c is not a cluster point of s. 00:03:16.580 --> 00:03:19.640 align:middle line:84% And you'll see that when c is not a cluster point of s, 00:03:19.640 --> 00:03:24.470 align:middle line:84% we'll have a silly situation for talking about continuity. 00:03:24.470 --> 00:03:33.200 align:middle line:84% So this is the following A theorem, which has three parts. 00:03:33.200 --> 00:03:38.420 align:middle line:90% So suppose s is a subset of r. 00:03:38.420 --> 00:03:42.150 align:middle line:90% c is an element of s. 00:03:42.150 --> 00:03:45.210 align:middle line:90% f is a function from s to r. 00:03:45.210 --> 00:03:57.030 align:middle line:84% So the first part is if c is not a cluster point of s, 00:03:57.030 --> 00:04:04.090 align:middle line:90% then f is continuous at c. 00:04:04.090 --> 00:04:07.180 align:middle line:84% So if we're looking at continuity at a point, 00:04:07.180 --> 00:04:08.920 align:middle line:84% the only interesting points to look at 00:04:08.920 --> 00:04:11.380 align:middle line:90% are cluster points of the set s. 00:04:11.380 --> 00:04:13.510 align:middle line:84% Otherwise, no matter what function it is, 00:04:13.510 --> 00:04:15.880 align:middle line:84% it will be continuous at such a point. 00:04:15.880 --> 00:04:19.230 align:middle line:90% 00:04:19.230 --> 00:04:20.720 align:middle line:90% The second is that-- 00:04:20.720 --> 00:04:23.510 align:middle line:84% now let's suppose we're in the more interesting case 00:04:23.510 --> 00:04:25.670 align:middle line:90% that c is a cluster point of s. 00:04:25.670 --> 00:04:38.990 align:middle line:84% Suppose c is a cluster point of s, 00:04:38.990 --> 00:04:42.260 align:middle line:84% which is essentially the only thing that doesn't-- 00:04:42.260 --> 00:04:44.780 align:middle line:84% that's missing from this definition and the definition 00:04:44.780 --> 00:04:46.100 align:middle line:90% for limit. 00:04:46.100 --> 00:04:50.090 align:middle line:90% Then f is continuous at c. 00:04:50.090 --> 00:04:52.460 align:middle line:84% And this is borne out in this theorem 00:04:52.460 --> 00:04:57.920 align:middle line:84% if and only if the limit as x goes to c of f of x 00:04:57.920 --> 00:04:59.000 align:middle line:90% equals f of c. 00:04:59.000 --> 00:05:04.930 align:middle line:90% 00:05:04.930 --> 00:05:07.250 align:middle line:84% And the third part of this theorem, 00:05:07.250 --> 00:05:10.510 align:middle line:84% which is now the analog of this theorem we proved here 00:05:10.510 --> 00:05:17.010 align:middle line:90% for limits, is the following-- 00:05:17.010 --> 00:05:30.720 align:middle line:84% f is continuous at c if and only if for every sequence 00:05:30.720 --> 00:05:45.080 align:middle line:84% xn of elements of s such that x of n converges to c, 00:05:45.080 --> 00:05:55.046 align:middle line:84% we have f of x sub n converges to f of c. 00:05:55.046 --> 00:05:59.140 align:middle line:90% 00:05:59.140 --> 00:06:01.810 align:middle line:84% So again, if c is not a cluster point of the set, 00:06:01.810 --> 00:06:04.400 align:middle line:84% then every function is going to be continuous at that point. 00:06:04.400 --> 00:06:07.630 align:middle line:90% So this is kind of a-- 00:06:07.630 --> 00:06:09.580 align:middle line:84% so if c is not a cluster point of s, 00:06:09.580 --> 00:06:12.160 align:middle line:90% this is a silly case to look at. 00:06:12.160 --> 00:06:14.970 align:middle line:90% 00:06:14.970 --> 00:06:15.470 align:middle line:90% All right. 00:06:15.470 --> 00:06:18.825 align:middle line:84% So let's prove the first theorem-- 00:06:18.825 --> 00:06:20.450 align:middle line:84% I mean, the first part of this theorem. 00:06:20.450 --> 00:06:25.300 align:middle line:90% 00:06:25.300 --> 00:06:29.360 align:middle line:90% So what's the intuition? 00:06:29.360 --> 00:06:31.100 align:middle line:84% Remember, continuity is a connection 00:06:31.100 --> 00:06:35.210 align:middle line:84% between f near a point and the function at the point. 00:06:35.210 --> 00:06:40.040 align:middle line:84% Now, if the only points near c is c, then 00:06:40.040 --> 00:06:46.250 align:middle line:84% f of x equals f of c for x near c because x can only be c. 00:06:46.250 --> 00:06:50.150 align:middle line:84% And therefore, that will be less than epsilon. 00:06:50.150 --> 00:06:55.370 align:middle line:84% So when we're at a point that's not a cluster point, 00:06:55.370 --> 00:06:57.450 align:middle line:84% it kind of removes the near part. 00:06:57.450 --> 00:07:02.105 align:middle line:84% And we're just looking at f at c and comparing f to itself at c. 00:07:02.105 --> 00:07:04.960 align:middle line:90% 00:07:04.960 --> 00:07:08.410 align:middle line:84% So let's suppose f is not a cluster point. 00:07:08.410 --> 00:07:11.358 align:middle line:90% 00:07:11.358 --> 00:07:13.150 align:middle line:84% So we want to prove continuity, so that's a 00:07:13.150 --> 00:07:15.070 align:middle line:90% for all epsilon delta argument. 00:07:15.070 --> 00:07:20.380 align:middle line:90% So let epsilon be positive. 00:07:20.380 --> 00:07:31.270 align:middle line:84% Since c is not a cluster point of s, what does this mean? 00:07:31.270 --> 00:07:33.180 align:middle line:84% So remember, to be a cluster point 00:07:33.180 --> 00:07:40.240 align:middle line:84% of a set s means for all delta positive, there exists-- 00:07:40.240 --> 00:07:44.490 align:middle line:84% so let me recall what it means to be a cluster 00:07:44.490 --> 00:07:47.160 align:middle line:90% point over here on the side. 00:07:47.160 --> 00:07:56.210 align:middle line:90% So c is a cluster point of s. 00:07:56.210 --> 00:08:02.690 align:middle line:84% This means for all delta positive, the set x minus delta 00:08:02.690 --> 00:08:09.070 align:middle line:84% x plus delta intersect c, intersect 00:08:09.070 --> 00:08:13.270 align:middle line:90% s takeaway c is not empty. 00:08:13.270 --> 00:08:16.910 align:middle line:90% 00:08:16.910 --> 00:08:18.950 align:middle line:84% So since c is not a cluster point, 00:08:18.950 --> 00:08:23.240 align:middle line:84% that means there exists some delta 0 so 00:08:23.240 --> 00:08:29.680 align:middle line:84% that this interval is disjoint from s takeaway c such 00:08:29.680 --> 00:08:33.309 align:middle line:84% that c minus delta 0, c plus delta 00:08:33.309 --> 00:08:41.159 align:middle line:84% 0 intersect s takeaway c equals the empty set. 00:08:41.159 --> 00:08:45.770 align:middle line:84% Now, if I include c in this intersection-- 00:08:45.770 --> 00:08:48.590 align:middle line:90% 00:08:48.590 --> 00:08:51.910 align:middle line:84% so another way of stating this is that the only thing-- 00:08:51.910 --> 00:08:55.750 align:middle line:84% since c is an element of s, the only thing 00:08:55.750 --> 00:08:59.740 align:middle line:84% that's in this intersection is c. 00:08:59.740 --> 00:09:06.490 align:middle line:90% 00:09:06.490 --> 00:09:13.440 align:middle line:84% So this here is the rigorous way of saying 00:09:13.440 --> 00:09:17.290 align:middle line:84% there's nothing in s near c except for c itself. 00:09:17.290 --> 00:09:21.870 align:middle line:84% And therefore, there's nothing to compare to f of c. 00:09:21.870 --> 00:09:24.690 align:middle line:84% So we'll choose delta to be this one because then there's 00:09:24.690 --> 00:09:25.380 align:middle line:90% nothing-- 00:09:25.380 --> 00:09:28.920 align:middle line:84% there's no x in s in this interval other than c. 00:09:28.920 --> 00:09:32.100 align:middle line:84% And then f of c minus f of c is 0, 00:09:32.100 --> 00:09:38.760 align:middle line:90% so choose delta to be delta 0. 00:09:38.760 --> 00:09:40.875 align:middle line:90% And if x minus-- 00:09:40.875 --> 00:09:46.170 align:middle line:90% 00:09:46.170 --> 00:09:51.160 align:middle line:84% so now we want to say that this delta works. 00:09:51.160 --> 00:09:55.800 align:middle line:84% So if x is in s and x minus c is less than delta, 00:09:55.800 --> 00:09:58.560 align:middle line:90% that implies x equals c. 00:09:58.560 --> 00:10:04.980 align:middle line:84% Because the only thing in this interval that's coming from s 00:10:04.980 --> 00:10:05.850 align:middle line:90% is c. 00:10:05.850 --> 00:10:09.750 align:middle line:84% And therefore, f of x minus f of c-- 00:10:09.750 --> 00:10:14.250 align:middle line:84% this is equal to f of c minus f of c 00:10:14.250 --> 00:10:16.155 align:middle line:84% equals 0, which is less than epsilon. 00:10:16.155 --> 00:10:20.180 align:middle line:90% 00:10:20.180 --> 00:10:23.510 align:middle line:84% So this is a really degenerate case 00:10:23.510 --> 00:10:26.300 align:middle line:84% of when you're trying to see if a function is 00:10:26.300 --> 00:10:27.770 align:middle line:90% continuous at a certain point. 00:10:27.770 --> 00:10:40.310 align:middle line:90% 00:10:40.310 --> 00:10:40.810 align:middle line:90% All right. 00:10:40.810 --> 00:10:43.390 align:middle line:84% So let's now move to the more interesting part 00:10:43.390 --> 00:10:44.740 align:middle line:90% that sees a cluster point. 00:10:44.740 --> 00:10:55.020 align:middle line:90% 00:10:55.020 --> 00:11:00.520 align:middle line:84% So suppose c is a cluster point of s. 00:11:00.520 --> 00:11:03.070 align:middle line:84% And we want to show f is continuous at c if and only 00:11:03.070 --> 00:11:07.480 align:middle line:84% if the limit of f as x goes to c of f of x equals f of c. 00:11:07.480 --> 00:11:09.560 align:middle line:84% Going this direction is really quite easy, 00:11:09.560 --> 00:11:12.670 align:middle line:84% so we're going to go this direction. 00:11:12.670 --> 00:11:15.940 align:middle line:84% And you should be able to prove this direction just 00:11:15.940 --> 00:11:18.020 align:middle line:90% from what I write down here. 00:11:18.020 --> 00:11:29.060 align:middle line:84% So suppose the limit as x goes to c of f of x equals f of c. 00:11:29.060 --> 00:11:32.340 align:middle line:84% I mean, this whole argument is kind of silly 00:11:32.340 --> 00:11:38.620 align:middle line:84% because the definitions are just so close except for now c 00:11:38.620 --> 00:11:40.120 align:middle line:84% is a cluster point, so that's not 00:11:40.120 --> 00:11:43.240 align:middle line:90% missing from either definition. 00:11:43.240 --> 00:11:47.680 align:middle line:84% But the only difference is this guy, you only look at x near c 00:11:47.680 --> 00:11:50.800 align:middle line:90% but not at c. 00:11:50.800 --> 00:11:53.830 align:middle line:84% So suppose the limit as x goes to c of f of x equals f of c. 00:11:53.830 --> 00:11:56.170 align:middle line:84% Now we want to show that f of x is-- 00:11:56.170 --> 00:12:01.360 align:middle line:84% that f is continuous at c, so let epsilon be positive. 00:12:01.360 --> 00:12:04.800 align:middle line:90% 00:12:04.800 --> 00:12:14.550 align:middle line:84% Since we have this, there exists a delta positive such 00:12:14.550 --> 00:12:30.350 align:middle line:84% that if x is in s and the absolute value of x minus c 00:12:30.350 --> 00:12:33.530 align:middle line:84% is bigger than 0 is less than delta 0, 00:12:33.530 --> 00:12:40.850 align:middle line:84% then f of x minus f of c is less than epsilon. 00:12:40.850 --> 00:12:43.070 align:middle line:84% And we'll just choose delta to be this delta 0. 00:12:43.070 --> 00:12:51.590 align:middle line:90% 00:12:51.590 --> 00:12:55.940 align:middle line:84% If x minus c is less than delta 0, then there are two cases. 00:12:55.940 --> 00:12:59.948 align:middle line:84% Either x is equal to c or x is not equal to z. 00:12:59.948 --> 00:13:00.448 align:middle line:90% See? 00:13:00.448 --> 00:13:04.440 align:middle line:90% 00:13:04.440 --> 00:13:05.820 align:middle line:90% So let's write it this way. 00:13:05.820 --> 00:13:18.030 align:middle line:90% 00:13:18.030 --> 00:13:25.460 align:middle line:84% If x equals c, then clearly, f of x minus f of c 00:13:25.460 --> 00:13:30.560 align:middle line:84% equals f of c minus f of c equals 0 is less than epsilon. 00:13:30.560 --> 00:13:34.230 align:middle line:90% 00:13:34.230 --> 00:13:42.730 align:middle line:84% And if x is not equal to c, then that's certainly bigger than 0, 00:13:42.730 --> 00:13:44.740 align:middle line:84% but we're still assuming it's less than delta, 00:13:44.740 --> 00:13:48.310 align:middle line:84% which remember, is this delta 0 coming from the fact 00:13:48.310 --> 00:13:52.390 align:middle line:84% that the limit as x goes to c of f of x-- 00:13:52.390 --> 00:13:53.560 align:middle line:90% I missed that there-- 00:13:53.560 --> 00:13:56.310 align:middle line:90% 00:13:56.310 --> 00:14:05.880 align:middle line:84% tells me that this is less than epsilon. 00:14:05.880 --> 00:14:12.122 align:middle line:84% So it just follows immediately from the definition. 00:14:12.122 --> 00:14:13.080 align:middle line:90% There's not much to it. 00:14:13.080 --> 00:14:18.490 align:middle line:90% 00:14:18.490 --> 00:14:18.990 align:middle line:90% OK. 00:14:18.990 --> 00:14:24.330 align:middle line:84% So now let's prove the third part of this theorem, which 00:14:24.330 --> 00:14:31.200 align:middle line:84% is an analog of that theorem we have over there for limits. 00:14:31.200 --> 00:14:38.240 align:middle line:90% 00:14:38.240 --> 00:14:41.780 align:middle line:90% So let's prove this direction. 00:14:41.780 --> 00:14:49.167 align:middle line:84% Let's suppose f is continuous at c, 00:14:49.167 --> 00:14:51.500 align:middle line:84% and now we want to prove this statement about sequences. 00:14:51.500 --> 00:14:53.900 align:middle line:90% So let me also-- 00:14:53.900 --> 00:14:55.610 align:middle line:84% when I prove the opposite direction, 00:14:55.610 --> 00:14:56.930 align:middle line:90% not have to label it. 00:14:56.930 --> 00:14:58.140 align:middle line:90% So let me put a star. 00:14:58.140 --> 00:15:01.730 align:middle line:84% This star is going to be the right-hand side 00:15:01.730 --> 00:15:02.690 align:middle line:90% of this statement. 00:15:02.690 --> 00:15:08.650 align:middle line:90% 00:15:08.650 --> 00:15:21.660 align:middle line:84% So let x sub n be a sequence in s of elements of s such 00:15:21.660 --> 00:15:23.430 align:middle line:90% that x converges to c. 00:15:23.430 --> 00:15:27.230 align:middle line:90% 00:15:27.230 --> 00:15:32.030 align:middle line:84% Now, I drew this picture for limits, 00:15:32.030 --> 00:15:34.145 align:middle line:84% and it's the same picture now for this guy. 00:15:34.145 --> 00:15:37.260 align:middle line:90% 00:15:37.260 --> 00:15:43.950 align:middle line:84% And we're wanting to show now that the limit as n 00:15:43.950 --> 00:15:49.890 align:middle line:84% goes to infinity of f of x sub n equals f of c. 00:15:49.890 --> 00:15:56.300 align:middle line:84% So here's the epsilon m argument, 00:15:56.300 --> 00:15:57.660 align:middle line:90% so let epsilon be positive. 00:15:57.660 --> 00:15:59.070 align:middle line:90% So now I'll go to the picture. 00:15:59.070 --> 00:16:01.040 align:middle line:90% Here's f of c. 00:16:01.040 --> 00:16:07.110 align:middle line:84% Here's f of c plus epsilon, f of c minus epsilon. 00:16:07.110 --> 00:16:08.640 align:middle line:90% So what do we know? 00:16:08.640 --> 00:16:11.660 align:middle line:84% We know the function is continuous at c. 00:16:11.660 --> 00:16:16.970 align:middle line:84% So that means that if I look at c, then 00:16:16.970 --> 00:16:27.840 align:middle line:84% there exists a delta so that if I'm in this interval here, 00:16:27.840 --> 00:16:31.710 align:middle line:84% everything in this interval gets mapped to inside 00:16:31.710 --> 00:16:32.520 align:middle line:90% of this interval. 00:16:32.520 --> 00:16:36.850 align:middle line:90% 00:16:36.850 --> 00:16:40.150 align:middle line:84% Now, what else do I know or what else am I assuming? 00:16:40.150 --> 00:16:42.340 align:middle line:84% x sub n is a sequence converging to s. 00:16:42.340 --> 00:16:50.690 align:middle line:84% So for large n, for n big enough for some capital M0, 00:16:50.690 --> 00:16:53.720 align:middle line:84% x sub n is going to lie inside this interval. 00:16:53.720 --> 00:16:57.080 align:middle line:84% If you like, in the definition of convergence of a sequence, 00:16:57.080 --> 00:16:59.540 align:middle line:84% epsilon is equal to delta in that definition. 00:16:59.540 --> 00:17:02.780 align:middle line:90% But don't mind that. 00:17:02.780 --> 00:17:06.540 align:middle line:84% So for all n big enough, x sub n lies in this interval. 00:17:06.540 --> 00:17:09.200 align:middle line:84% And since this interval gets mapped 00:17:09.200 --> 00:17:14.000 align:middle line:84% inside this interval by f, this f of x 00:17:14.000 --> 00:17:17.930 align:middle line:84% sub n will end up in this interval within epsilon 00:17:17.930 --> 00:17:22.050 align:middle line:84% to f of c as long as n is big enough. 00:17:22.050 --> 00:17:25.650 align:middle line:84% And so this picture I just went through, this is the proof. 00:17:25.650 --> 00:17:27.240 align:middle line:84% And now we just need to write it out. 00:17:27.240 --> 00:17:39.440 align:middle line:90% 00:17:39.440 --> 00:17:44.190 align:middle line:84% Since f is continuous at c, there 00:17:44.190 --> 00:17:54.690 align:middle line:84% exists a delta positive such that if x minus c is 00:17:54.690 --> 00:17:59.640 align:middle line:84% less than delta, this tells me f of x minus f 00:17:59.640 --> 00:18:01.755 align:middle line:90% of c is less than epsilon. 00:18:01.755 --> 00:18:05.960 align:middle line:90% 00:18:05.960 --> 00:18:10.880 align:middle line:84% Now, since the sequence x sub n converges to c, 00:18:10.880 --> 00:18:14.540 align:middle line:84% there exists some integer capital M0 00:18:14.540 --> 00:18:22.100 align:middle line:84% such that for all n bigger than or equal to 0, x 00:18:22.100 --> 00:18:24.440 align:middle line:84% sub n minus c is less than delta. 00:18:24.440 --> 00:18:27.470 align:middle line:90% 00:18:27.470 --> 00:18:30.290 align:middle line:84% And therefore, for all n bigger than or equal to m sub 0, 00:18:30.290 --> 00:18:36.170 align:middle line:84% f of x in will be within epsilon of f of c. 00:18:36.170 --> 00:18:38.315 align:middle line:90% So this will be the m we choose. 00:18:38.315 --> 00:18:40.960 align:middle line:90% 00:18:40.960 --> 00:18:47.070 align:middle line:84% Then if n is bigger than or equal to M, 00:18:47.070 --> 00:18:53.490 align:middle line:84% we get that x sub n minus x minus c is less than delta, 00:18:53.490 --> 00:18:57.570 align:middle line:84% which implies by how delta's defined-- 00:18:57.570 --> 00:19:01.740 align:middle line:84% if x minus c is within distance delta, f of x minus f of c 00:19:01.740 --> 00:19:04.170 align:middle line:90% is within distance epsilon. 00:19:04.170 --> 00:19:13.580 align:middle line:84% We get f of x sub n minus f of c is less than epsilon. 00:19:13.580 --> 00:19:18.690 align:middle line:90% 00:19:18.690 --> 00:19:20.190 align:middle line:90% So that gives us one direction. 00:19:20.190 --> 00:19:29.650 align:middle line:90% 00:19:29.650 --> 00:19:30.150 align:middle line:90% All right. 00:19:30.150 --> 00:19:32.830 align:middle line:84% So now we'll prove the opposite direction. 00:19:32.830 --> 00:19:44.530 align:middle line:84% So we're assuming that this statement star holds, namely 00:19:44.530 --> 00:19:46.630 align:middle line:84% for all sequence x sub n's converging 00:19:46.630 --> 00:19:50.200 align:middle line:84% to c, we have f of x sub n converging to f of c. 00:19:50.200 --> 00:19:55.810 align:middle line:84% And we want to prove that the limit or that f 00:19:55.810 --> 00:19:57.040 align:middle line:90% is continuous at c. 00:19:57.040 --> 00:20:02.780 align:middle line:90% 00:20:02.780 --> 00:20:05.090 align:middle line:84% And we're going to do this by contradiction, which 00:20:05.090 --> 00:20:07.820 align:middle line:84% is the same way we proved the opposite direction 00:20:07.820 --> 00:20:10.460 align:middle line:90% for the theorem about limits. 00:20:10.460 --> 00:20:18.320 align:middle line:84% So the proof is by contradiction, namely 00:20:18.320 --> 00:20:20.730 align:middle line:84% suppose the conclusion we want does not hold. 00:20:20.730 --> 00:20:30.110 align:middle line:84% So suppose f is not continuous at c. 00:20:30.110 --> 00:20:34.160 align:middle line:84% So let me recall we negated that definition in the last lecture, 00:20:34.160 --> 00:20:35.660 align:middle line:90% but we'll negate it again. 00:20:35.660 --> 00:20:40.710 align:middle line:84% For this, this means that there exists a bad epsilon 00:20:40.710 --> 00:20:46.350 align:middle line:84% so that for all delta positive, there exists an x 00:20:46.350 --> 00:20:51.570 align:middle line:84% and s satisfying x minus c is less than delta. 00:20:51.570 --> 00:21:00.570 align:middle line:84% And f of x minus f of c is bigger than or equal 00:21:00.570 --> 00:21:01.620 align:middle line:90% to epsilon 0. 00:21:01.620 --> 00:21:11.162 align:middle line:90% 00:21:11.162 --> 00:21:11.750 align:middle line:90% All right. 00:21:11.750 --> 00:21:16.220 align:middle line:84% So since we're assuming f is not continuous 00:21:16.220 --> 00:21:20.490 align:middle line:84% or exists at epsilon 0, so we have all this, 00:21:20.490 --> 00:21:22.650 align:middle line:90% which holds for every delta. 00:21:22.650 --> 00:21:26.550 align:middle line:84% And now we'll choose delta to be 1, 1/2, 1/3, and so on. 00:21:26.550 --> 00:21:29.520 align:middle line:90% 00:21:29.520 --> 00:21:34.050 align:middle line:84% Then there exists x sub 1 in s such 00:21:34.050 --> 00:21:38.670 align:middle line:84% that x sub 1 minus c is less than 1 00:21:38.670 --> 00:21:45.630 align:middle line:84% and f of x sub 1 minus f of c is bigger than or equal to epsilon 00:21:45.630 --> 00:21:46.170 align:middle line:90% 0. 00:21:46.170 --> 00:21:47.160 align:middle line:90% That's just delta. 00:21:47.160 --> 00:21:48.750 align:middle line:90% If you like delta, equals 1. 00:21:48.750 --> 00:21:53.670 align:middle line:90% 00:21:53.670 --> 00:21:54.705 align:middle line:90% And now we continue. 00:21:54.705 --> 00:21:57.585 align:middle line:90% 00:21:57.585 --> 00:22:01.170 align:middle line:84% There exists x sub 2 in this such 00:22:01.170 --> 00:22:05.490 align:middle line:84% that x sub 2 minus c is less than 1/2. 00:22:05.490 --> 00:22:09.510 align:middle line:84% And so here, this is delta equals 1/2. 00:22:09.510 --> 00:22:14.160 align:middle line:90% 00:22:14.160 --> 00:22:17.250 align:middle line:84% So this is delta equals 1/2, and so on. 00:22:17.250 --> 00:22:18.540 align:middle line:90% So then we conclude-- 00:22:18.540 --> 00:22:22.590 align:middle line:90% 00:22:22.590 --> 00:22:26.640 align:middle line:84% so if you like, omit this from the proof. 00:22:26.640 --> 00:22:30.810 align:middle line:84% And this is really what's behind the next statement 00:22:30.810 --> 00:22:32.220 align:middle line:90% I'm about to make. 00:22:32.220 --> 00:22:36.180 align:middle line:84% Then for all natural numbers n, there 00:22:36.180 --> 00:22:46.560 align:middle line:84% exists an x sub n in s such that x sub n minus c is less than 1 00:22:46.560 --> 00:22:53.760 align:middle line:84% over n, and f of x sub n minus f of c 00:22:53.760 --> 00:22:55.620 align:middle line:84% is bigger than or equal to epsilon 0. 00:22:55.620 --> 00:23:00.770 align:middle line:90% 00:23:00.770 --> 00:23:03.200 align:middle line:84% Now let's take a look at this sequence. 00:23:03.200 --> 00:23:05.480 align:middle line:90% We're trying to break something. 00:23:05.480 --> 00:23:08.990 align:middle line:84% And we'll end up breaking the star assumption. 00:23:08.990 --> 00:23:11.630 align:middle line:84% So now we have this sequence, x sub n 00:23:11.630 --> 00:23:16.610 align:middle line:84% of s, which we're getting closer and closer to c. 00:23:16.610 --> 00:23:20.810 align:middle line:84% So we have 0 is bigger than or equal to x sub n minus 00:23:20.810 --> 00:23:22.100 align:middle line:90% c is bigger-- 00:23:22.100 --> 00:23:23.480 align:middle line:90% is less than 1 over n. 00:23:23.480 --> 00:23:25.820 align:middle line:90% So this converges to 0. 00:23:25.820 --> 00:23:27.620 align:middle line:90% This converges to 0. 00:23:27.620 --> 00:23:29.810 align:middle line:84% So by the squeeze theorem, the absolute value 00:23:29.810 --> 00:23:33.200 align:middle line:84% of x sub n minus c converges to 0. 00:23:33.200 --> 00:23:38.110 align:middle line:84% And therefore, x sub n converges to c. 00:23:38.110 --> 00:23:44.230 align:middle line:84% By squeeze theorem, we get that xn converges to c. 00:23:44.230 --> 00:23:46.360 align:middle line:84% Now, since we're assuming star holds-- 00:23:46.360 --> 00:23:52.540 align:middle line:84% that's the right side of 3, of the if and only if-- 00:23:52.540 --> 00:23:56.340 align:middle line:84% it must be the case that f of xn converges to f of c. 00:23:56.340 --> 00:24:11.570 align:middle line:90% 00:24:11.570 --> 00:24:13.430 align:middle line:84% That's what star tells me-- that if I 00:24:13.430 --> 00:24:17.570 align:middle line:84% take a sequence converging to c, f of x converges to f of c. 00:24:17.570 --> 00:24:21.250 align:middle line:84% But each one of these, remember, is 00:24:21.250 --> 00:24:31.830 align:middle line:84% bigger than or equal to epsilon 0, which is a contradiction. 00:24:31.830 --> 00:24:33.375 align:middle line:90% Epsilon 0 is a positive number. 00:24:33.375 --> 00:24:36.980 align:middle line:90% 00:24:36.980 --> 00:24:43.930 align:middle line:84% And that concludes the proof of this theorem. 00:24:43.930 --> 00:24:46.240 align:middle line:84% That gives us an equivalent way of stating continuity 00:24:46.240 --> 00:24:47.320 align:middle line:90% in terms of sequences. 00:24:47.320 --> 00:24:52.660 align:middle line:84% And just like for limits, this will 00:24:52.660 --> 00:24:57.580 align:middle line:84% allow us to use what we know about sequences to conclude 00:24:57.580 --> 00:25:00.430 align:middle line:84% analogous facts for continuous functions. 00:25:00.430 --> 00:25:15.340 align:middle line:90% 00:25:15.340 --> 00:25:17.800 align:middle line:84% So let's look at a non-trivial example 00:25:17.800 --> 00:25:22.570 align:middle line:84% of a continuous function and also put this theorem 00:25:22.570 --> 00:25:26.830 align:middle line:90% to use, this previous theorem. 00:25:26.830 --> 00:25:38.800 align:middle line:84% The functions f of x equals sine x and g of x equals cosine x, 00:25:38.800 --> 00:25:40.420 align:middle line:90% these are continuous functions. 00:25:40.420 --> 00:25:50.720 align:middle line:84% So meaning their domain is the set of real numbers, 00:25:50.720 --> 00:25:53.320 align:middle line:84% so they're continuous at every real number. 00:25:53.320 --> 00:25:56.020 align:middle line:84% They're continuous at c for every c, a real number. 00:25:56.020 --> 00:26:00.680 align:middle line:90% 00:26:00.680 --> 00:26:03.050 align:middle line:90% So we're going to use this-- 00:26:03.050 --> 00:26:07.220 align:middle line:90% 00:26:07.220 --> 00:26:09.310 align:middle line:84% well, we're not going to use this theorem just yet 00:26:09.310 --> 00:26:10.010 align:middle line:90% on this part. 00:26:10.010 --> 00:26:12.385 align:middle line:84% We're actually going to prove sine is continuous directly 00:26:12.385 --> 00:26:17.590 align:middle line:84% from the epsilon delta definition, which is always 00:26:17.590 --> 00:26:18.090 align:middle line:90% good. 00:26:18.090 --> 00:26:27.360 align:middle line:90% So first, we claim sine x. 00:26:27.360 --> 00:26:29.790 align:middle line:84% So before I say what we're going to do, 00:26:29.790 --> 00:26:31.380 align:middle line:84% let me just give you a quick refresher 00:26:31.380 --> 00:26:34.350 align:middle line:84% on what you can prove just from the definition of sine 00:26:34.350 --> 00:26:34.860 align:middle line:90% and cosine. 00:26:34.860 --> 00:26:37.860 align:middle line:90% Remember, I'm not-- 00:26:37.860 --> 00:26:42.750 align:middle line:90% 00:26:42.750 --> 00:26:46.470 align:middle line:84% I would usually put this to the class and see who remembers 00:26:46.470 --> 00:26:47.520 align:middle line:90% and who doesn't. 00:26:47.520 --> 00:26:50.040 align:middle line:84% So this is also supposed to be a unit circle, although it 00:26:50.040 --> 00:26:52.290 align:middle line:90% doesn't quite look like that. 00:26:52.290 --> 00:26:58.170 align:middle line:84% Remember that sine and cosine are defined as you travel along 00:26:58.170 --> 00:27:01.440 align:middle line:84% signed distance x along the circle, 00:27:01.440 --> 00:27:12.140 align:middle line:84% and you arrive at a point on the unit circle, which you-- 00:27:12.140 --> 00:27:13.910 align:middle line:90% so it's an ordered pair. 00:27:13.910 --> 00:27:16.340 align:middle line:84% The first element you call cosine x. 00:27:16.340 --> 00:27:18.050 align:middle line:84% The second element you call sine x. 00:27:18.050 --> 00:27:20.910 align:middle line:84% So that's how sine of x is defined. 00:27:20.910 --> 00:27:24.050 align:middle line:90% So now simply-- 00:27:24.050 --> 00:27:29.180 align:middle line:84% I'm not going to do this because this is trig, not analysis. 00:27:29.180 --> 00:27:37.770 align:middle line:84% But from the definition of sine and cosine 00:27:37.770 --> 00:27:53.930 align:middle line:84% from the unit circle, we have that, of course, 00:27:53.930 --> 00:28:00.980 align:middle line:84% all x and r sine x plus cosine squared x is equal to 1. 00:28:00.980 --> 00:28:04.040 align:middle line:84% And therefore, each of these individual things 00:28:04.040 --> 00:28:06.200 align:middle line:84% has to be less than or equal to 1, which upon 00:28:06.200 --> 00:28:08.345 align:middle line:90% taking square roots means that-- 00:28:08.345 --> 00:28:20.130 align:middle line:90% 00:28:20.130 --> 00:28:21.870 align:middle line:90% so you have these. 00:28:21.870 --> 00:28:26.130 align:middle line:84% Now, you can also make a better estimate for sine x 00:28:26.130 --> 00:28:31.800 align:middle line:84% when x is close to 0, which is only useful for x close to 0, 00:28:31.800 --> 00:28:34.980 align:middle line:84% but which is the following-- that for all x, 00:28:34.980 --> 00:28:41.370 align:middle line:84% sine x is less than or equal to the absolute value of x, just 00:28:41.370 --> 00:28:48.310 align:middle line:84% obtained by comparing the length of one side of a triangle 00:28:48.310 --> 00:28:50.305 align:middle line:84% from this picture with the length of the arc. 00:28:50.305 --> 00:28:54.382 align:middle line:90% 00:28:54.382 --> 00:28:55.840 align:middle line:84% So you have this, and then you also 00:28:55.840 --> 00:29:06.560 align:middle line:84% have the angle sum formula, which 00:29:06.560 --> 00:29:14.450 align:middle line:84% says that sine of a plus b equals sine a cosine b 00:29:14.450 --> 00:29:18.770 align:middle line:90% plus cosine a sine b. 00:29:18.770 --> 00:29:21.640 align:middle line:90% 00:29:21.640 --> 00:29:25.930 align:middle line:90% And you also have the-- 00:29:25.930 --> 00:29:29.560 align:middle line:84% I can't remember the name of the exact formula. 00:29:29.560 --> 00:29:34.330 align:middle line:84% I think it's difference to product or something of that 00:29:34.330 --> 00:29:43.060 align:middle line:84% nature, which says sine of a minus sine a b, you can write-- 00:29:43.060 --> 00:29:45.700 align:middle line:84% so this is using the previous formula-- 00:29:45.700 --> 00:29:53.380 align:middle line:84% twice sine of a minus b over 2 cosine of a plus b over 2. 00:29:53.380 --> 00:29:59.370 align:middle line:90% 00:29:59.370 --> 00:30:02.850 align:middle line:84% So using these elementary properties of sine and cosine, 00:30:02.850 --> 00:30:06.660 align:middle line:84% we'll now show that sine of x is continuous. 00:30:06.660 --> 00:30:12.080 align:middle line:90% 00:30:12.080 --> 00:30:23.200 align:middle line:84% So first thing we want to show is that sine x 00:30:23.200 --> 00:30:26.605 align:middle line:90% is a continuous function. 00:30:26.605 --> 00:30:29.780 align:middle line:90% 00:30:29.780 --> 00:30:30.580 align:middle line:90% So let's see. 00:30:30.580 --> 00:30:35.120 align:middle line:84% In R, we're going to show sine is continuous at c. 00:30:35.120 --> 00:30:38.190 align:middle line:90% Let epsilon be positive. 00:30:38.190 --> 00:30:40.757 align:middle line:84% And now we have to say how to choose delta 00:30:40.757 --> 00:30:41.840 align:middle line:90% depending on this epsilon. 00:30:41.840 --> 00:30:46.470 align:middle line:90% 00:30:46.470 --> 00:30:47.670 align:middle line:90% Choose delta to be epsilon. 00:30:47.670 --> 00:30:52.732 align:middle line:90% 00:30:52.732 --> 00:30:54.440 align:middle line:84% Now we'll use these elementary properties 00:30:54.440 --> 00:30:56.420 align:middle line:90% to show that this delta works. 00:30:56.420 --> 00:30:58.950 align:middle line:90% 00:30:58.950 --> 00:31:05.980 align:middle line:84% Then if x minus c is less than delta, 00:31:05.980 --> 00:31:10.540 align:middle line:84% we now want to show that f of x minus f of c, sine x minus sine 00:31:10.540 --> 00:31:12.230 align:middle line:90% c, is less than epsilon. 00:31:12.230 --> 00:31:15.400 align:middle line:84% So sine x minus sine c, this is equal to-- 00:31:15.400 --> 00:31:23.520 align:middle line:84% by this last formula we have on the board over here, 00:31:23.520 --> 00:31:29.310 align:middle line:84% this is equal to twice sine x minus c 00:31:29.310 --> 00:31:35.550 align:middle line:90% over 2 cosine x plus c over 2. 00:31:35.550 --> 00:31:38.940 align:middle line:84% And now this is the product of 2 times the absolute value 00:31:38.940 --> 00:31:41.520 align:middle line:84% of sine times absolute value of cosine. 00:31:41.520 --> 00:31:43.560 align:middle line:84% By the second property over here, 00:31:43.560 --> 00:31:46.140 align:middle line:84% cosine is always bounded by 1, so this is less than 00:31:46.140 --> 00:31:53.660 align:middle line:84% or equal to 2 times the absolute value of sine x minus c over 2. 00:31:53.660 --> 00:31:56.270 align:middle line:84% And now we use this third property here 00:31:56.270 --> 00:31:59.815 align:middle line:84% that sine x is less than or equal to the absolute value 00:31:59.815 --> 00:32:01.190 align:middle line:84% of that thing you're sticking in. 00:32:01.190 --> 00:32:06.800 align:middle line:84% So this is less than or equal to 2 times x minus c over 2, 00:32:06.800 --> 00:32:09.260 align:middle line:90% which equals x minus c. 00:32:09.260 --> 00:32:10.670 align:middle line:90% So now we're in business. 00:32:10.670 --> 00:32:14.450 align:middle line:84% We've connected f of x to x minus c. 00:32:14.450 --> 00:32:16.430 align:middle line:84% So this is less than or equal to-- so this 00:32:16.430 --> 00:32:21.650 align:middle line:84% is less than delta, which equals epsilon by our choice of delta. 00:32:21.650 --> 00:32:24.875 align:middle line:84% And therefore, sine x minus sine c is less than epsilon. 00:32:24.875 --> 00:32:30.620 align:middle line:90% 00:32:30.620 --> 00:32:34.980 align:middle line:84% Sine x minus sine c is less than epsilon. 00:32:34.980 --> 00:32:45.920 align:middle line:84% So thus, function sine x is continuous at c. 00:32:45.920 --> 00:32:52.530 align:middle line:90% 00:32:52.530 --> 00:32:58.920 align:middle line:84% Now we'll use this and the theorem we showed a minute ago. 00:32:58.920 --> 00:33:01.920 align:middle line:90% 00:33:01.920 --> 00:33:03.000 align:middle line:90% Have I erased it? 00:33:03.000 --> 00:33:04.830 align:middle line:84% Yes, I have now officially erased it. 00:33:04.830 --> 00:33:06.690 align:middle line:84% Anyways, we used the previous theorem 00:33:06.690 --> 00:33:09.840 align:middle line:84% that we proved for the equivalence of continuity 00:33:09.840 --> 00:33:14.850 align:middle line:84% and convergence of sequences to show that cosine is continuous. 00:33:14.850 --> 00:33:19.720 align:middle line:90% 00:33:19.720 --> 00:33:31.310 align:middle line:84% So let c be an element of R. Let xn 00:33:31.310 --> 00:33:39.070 align:middle line:90% be a sequence converging to c. 00:33:39.070 --> 00:33:50.920 align:middle line:84% And we now want to show that cosine of x sub n 00:33:50.920 --> 00:33:55.612 align:middle line:90% converges to cosine of c. 00:33:55.612 --> 00:33:57.320 align:middle line:84% Once we've done that, then by the theorem 00:33:57.320 --> 00:34:00.910 align:middle line:84% we proved previously, we can conclude 00:34:00.910 --> 00:34:02.650 align:middle line:90% that cosine is continuous at c. 00:34:02.650 --> 00:34:09.598 align:middle line:90% 00:34:09.598 --> 00:34:10.515 align:middle line:90% Now, here's the thing. 00:34:10.515 --> 00:34:13.595 align:middle line:90% 00:34:13.595 --> 00:34:23.790 align:middle line:84% For all x and r, we have cosine of x 00:34:23.790 --> 00:34:29.010 align:middle line:84% is equal to sine of x plus pi over 2. 00:34:29.010 --> 00:34:31.545 align:middle line:84% We can deduce that simply from this angle sum formula. 00:34:31.545 --> 00:34:34.850 align:middle line:90% 00:34:34.850 --> 00:34:37.280 align:middle line:84% Take a equals c, b equals pi over 2. 00:34:37.280 --> 00:34:40.870 align:middle line:90% 00:34:40.870 --> 00:34:43.000 align:middle line:84% I'm using that cosine of pi over 2 is 0. 00:34:43.000 --> 00:34:45.699 align:middle line:90% 00:34:45.699 --> 00:34:50.420 align:middle line:84% And this is what we'll use, and the fact that we know sine 00:34:50.420 --> 00:34:51.530 align:middle line:90% is continuous. 00:34:51.530 --> 00:34:54.949 align:middle line:84% Since the sequence xn converges to c, 00:34:54.949 --> 00:35:01.700 align:middle line:84% this implies the sequence x sub n plus pi over 2 converges 00:35:01.700 --> 00:35:03.320 align:middle line:90% to c plus pi over 2. 00:35:03.320 --> 00:35:06.000 align:middle line:90% 00:35:06.000 --> 00:35:11.390 align:middle line:84% Now, this sequence x sub n plus pi over 2 converges 00:35:11.390 --> 00:35:12.450 align:middle line:90% to this number. 00:35:12.450 --> 00:35:15.080 align:middle line:84% And since sine is continuous, we have 00:35:15.080 --> 00:35:21.740 align:middle line:84% by the previous theorem that sine of this converges to this. 00:35:21.740 --> 00:35:24.830 align:middle line:90% And therefore, cosine of-- 00:35:24.830 --> 00:35:29.331 align:middle line:90% 00:35:29.331 --> 00:35:36.530 align:middle line:84% cosine of x sub n, which is equal to sine of x sub n 00:35:36.530 --> 00:35:43.760 align:middle line:84% plus pi over 2, converges to sine of c plus pi over 2, which 00:35:43.760 --> 00:35:48.735 align:middle line:90% equals cosine of c, i.e. 00:35:48.735 --> 00:35:56.230 align:middle line:90% 00:35:56.230 --> 00:35:58.600 align:middle line:84% cosine of x sub n converges to cosine of c. 00:35:58.600 --> 00:36:03.100 align:middle line:84% And therefore, we've now shown that cosine is also continuous. 00:36:03.100 --> 00:36:17.140 align:middle line:90% 00:36:17.140 --> 00:36:19.040 align:middle line:84% So let's answer this question real quick. 00:36:19.040 --> 00:36:22.570 align:middle line:90% So the answer is no. 00:36:22.570 --> 00:36:27.340 align:middle line:84% You can find a function which is discontinuous 00:36:27.340 --> 00:36:29.245 align:middle line:84% at every point, every real number. 00:36:29.245 --> 00:36:34.240 align:middle line:90% 00:36:34.240 --> 00:36:36.418 align:middle line:84% So maybe this is an example rather than a theorem, 00:36:36.418 --> 00:36:37.710 align:middle line:90% but I'll state it as a theorem. 00:36:37.710 --> 00:36:41.490 align:middle line:90% 00:36:41.490 --> 00:36:44.210 align:middle line:84% Let f of x be the function which takes the value 00:36:44.210 --> 00:36:56.220 align:middle line:84% 1 if x is irrational, 0 if x is not irrational. 00:36:56.220 --> 00:37:02.170 align:middle line:90% 00:37:02.170 --> 00:37:17.180 align:middle line:84% Then f is not continuous at every c and r. 00:37:17.180 --> 00:37:21.120 align:middle line:84% So first off, don't try to plot this 00:37:21.120 --> 00:37:23.370 align:middle line:84% using Matlab or Mathematica or anything 00:37:23.370 --> 00:37:26.190 align:middle line:84% because first off, computers can't 00:37:26.190 --> 00:37:28.270 align:middle line:90% deal with irrational numbers. 00:37:28.270 --> 00:37:32.220 align:middle line:84% So you will just get 1 no matter what you stick into f. 00:37:32.220 --> 00:37:36.970 align:middle line:90% 00:37:36.970 --> 00:37:37.470 align:middle line:90% OK. 00:37:37.470 --> 00:37:40.200 align:middle line:84% So you have this function, which is 1 if x is a rational 0. 00:37:40.200 --> 00:37:44.100 align:middle line:84% If x is not a rational number, if it's irrational, 00:37:44.100 --> 00:37:46.262 align:middle line:84% let's show it's discontinuous at every c. 00:37:46.262 --> 00:37:49.160 align:middle line:90% 00:37:49.160 --> 00:37:54.080 align:middle line:84% A different way, if you like, to state the theorem which 00:37:54.080 --> 00:37:54.965 align:middle line:90% I've already erased-- 00:37:54.965 --> 00:37:58.310 align:middle line:90% 00:37:58.310 --> 00:38:00.650 align:middle line:84% but let me just state it slightly differently. 00:38:00.650 --> 00:38:04.510 align:middle line:90% 00:38:04.510 --> 00:38:09.450 align:middle line:90% f is not continuous at c. 00:38:09.450 --> 00:38:13.398 align:middle line:84% So remember, the two statements are equivalent if 00:38:13.398 --> 00:38:15.190 align:middle line:84% and only if their negations are equivalent. 00:38:15.190 --> 00:38:19.500 align:middle line:84% So f is not continuous at c if and only 00:38:19.500 --> 00:38:33.310 align:middle line:84% if there exists a sequence x sub n such that xn 00:38:33.310 --> 00:38:38.480 align:middle line:90% converges to c and f of xn-- 00:38:38.480 --> 00:38:41.820 align:middle line:90% 00:38:41.820 --> 00:38:51.390 align:middle line:84% the sequence fn of xn does not converge to f of c. 00:38:51.390 --> 00:38:53.460 align:middle line:84% So maybe it just doesn't converge at all, 00:38:53.460 --> 00:38:56.833 align:middle line:84% or maybe it converges to something other than f of c. 00:38:56.833 --> 00:38:59.250 align:middle line:84% So this is just a restatement of the theorem we've already 00:38:59.250 --> 00:39:03.150 align:middle line:84% proven but now using negations of the statements that 00:39:03.150 --> 00:39:06.650 align:middle line:90% appeared on the if and only if. 00:39:06.650 --> 00:39:10.010 align:middle line:84% So there's two cases to consider for this. 00:39:10.010 --> 00:39:11.070 align:middle line:90% So now we're going-- 00:39:11.070 --> 00:39:15.680 align:middle line:84% so this is a statement of the theorem that we're-- 00:39:15.680 --> 00:39:17.630 align:middle line:84% a restatement of the theorem we're using. 00:39:17.630 --> 00:39:20.910 align:middle line:84% And now we're going to prove the theorem that I've stated here. 00:39:20.910 --> 00:39:23.540 align:middle line:84% So let c be an element of R. We're 00:39:23.540 --> 00:39:26.690 align:middle line:84% going to show that this function is discontinuous at c. 00:39:26.690 --> 00:39:30.110 align:middle line:84% There are two cases to consider, c is a rational number 00:39:30.110 --> 00:39:33.420 align:middle line:90% or c is an irrational number. 00:39:33.420 --> 00:39:40.430 align:middle line:84% So case 1-- c is a rational number. 00:39:40.430 --> 00:39:42.440 align:middle line:84% Let's show f is not continuous at c. 00:39:42.440 --> 00:39:50.870 align:middle line:90% 00:39:50.870 --> 00:39:57.690 align:middle line:84% Now, we know that for every natural number n, 00:39:57.690 --> 00:40:07.460 align:middle line:84% there exists an element x sub n and the complement of q, 00:40:07.460 --> 00:40:13.160 align:middle line:84% so an irrational number, such that c is less than x sub n 00:40:13.160 --> 00:40:15.650 align:middle line:90% is less than c plus 1 over n. 00:40:15.650 --> 00:40:17.750 align:middle line:84% So we proved this in an assignment. 00:40:17.750 --> 00:40:19.850 align:middle line:84% Between any two real numbers, I can 00:40:19.850 --> 00:40:22.190 align:middle line:84% find a rational number, an irrational number 00:40:22.190 --> 00:40:24.040 align:middle line:90% in between them. 00:40:24.040 --> 00:40:27.640 align:middle line:84% So that's what this statement says, is for each n, 00:40:27.640 --> 00:40:30.550 align:middle line:84% I can find an irrational number between the number c 00:40:30.550 --> 00:40:32.510 align:middle line:90% and the number c plus 1 over n. 00:40:32.510 --> 00:40:37.450 align:middle line:90% 00:40:37.450 --> 00:40:41.480 align:middle line:84% Now, by the squeeze theorem, this 00:40:41.480 --> 00:40:44.750 align:middle line:84% is just a constant sequence, if you like, converging to c. 00:40:44.750 --> 00:40:45.980 align:middle line:90% This convergence to c. 00:40:45.980 --> 00:40:48.740 align:middle line:84% And as n goes to infinity, this convergence to c. 00:40:48.740 --> 00:40:54.660 align:middle line:84% So I get that the sequence x sub n converges to c. 00:40:54.660 --> 00:41:00.500 align:middle line:84% This will be my bad sequence to satisfy this conclusion 00:41:00.500 --> 00:41:01.000 align:middle line:90% because-- 00:41:01.000 --> 00:41:07.750 align:middle line:90% 00:41:07.750 --> 00:41:16.424 align:middle line:84% so xn converges to c, but if I look at f of xn, 00:41:16.424 --> 00:41:21.440 align:middle line:84% if I take the limit as n goes to infinity of f of xn, 00:41:21.440 --> 00:41:24.120 align:middle line:84% since all the xn's are not rational-- 00:41:24.120 --> 00:41:25.730 align:middle line:90% they're irrational numbers-- 00:41:25.730 --> 00:41:27.965 align:middle line:84% and I stick them into f, I get 0. 00:41:27.965 --> 00:41:32.940 align:middle line:90% 00:41:32.940 --> 00:41:35.890 align:middle line:84% And this does not equal 1, which equals 00:41:35.890 --> 00:41:40.530 align:middle line:84% f of c because we're in the case that c is a rational number. 00:41:40.530 --> 00:41:43.740 align:middle line:84% So here we use the density of the irrationals 00:41:43.740 --> 00:41:45.600 align:middle line:84% to show that this function is not 00:41:45.600 --> 00:41:49.390 align:middle line:84% continuous at a rational number, and we'll 00:41:49.390 --> 00:42:01.790 align:middle line:84% do the same thing for the case that c is an irrational number. 00:42:01.790 --> 00:42:03.740 align:middle line:84% It's the same proof, except now we 00:42:03.740 --> 00:42:05.120 align:middle line:90% essentially take compliments. 00:42:05.120 --> 00:42:07.860 align:middle line:90% 00:42:07.860 --> 00:42:11.130 align:middle line:84% We proved this theorem right after the Archimedean property 00:42:11.130 --> 00:42:13.470 align:middle line:84% of r that for any two real numbers, 00:42:13.470 --> 00:42:15.940 align:middle line:84% I can find a rational number in between them. 00:42:15.940 --> 00:42:23.820 align:middle line:84% So for every n, there exists a rational number such 00:42:23.820 --> 00:42:31.518 align:middle line:84% that c is less than x of n is less in c plus 1 over n. 00:42:31.518 --> 00:42:36.990 align:middle line:90% 00:42:36.990 --> 00:42:41.710 align:middle line:84% And again, by the squeeze theorem, 00:42:41.710 --> 00:42:44.050 align:middle line:84% it follows that xn converges to c. 00:42:44.050 --> 00:42:46.760 align:middle line:90% 00:42:46.760 --> 00:42:49.910 align:middle line:84% And if we look at the limit as n goes 00:42:49.910 --> 00:42:53.510 align:middle line:84% to infinity of f of x of n, this is 00:42:53.510 --> 00:42:59.960 align:middle line:84% equal to-- all of these values are 1, which does not 00:42:59.960 --> 00:43:02.780 align:middle line:90% equal 0, which equals f of c. 00:43:02.780 --> 00:43:06.770 align:middle line:84% And c is assumed to be an irrational number. 00:43:06.770 --> 00:43:11.860 align:middle line:84% So this function is not continuous 00:43:11.860 --> 00:43:14.750 align:middle line:90% at every real number. 00:43:14.750 --> 00:43:17.542 align:middle line:90% So just terminology here. 00:43:17.542 --> 00:43:19.000 align:middle line:84% If you hear me say this, when I say 00:43:19.000 --> 00:43:26.080 align:middle line:84% something is not continuous, I'll often say discontinuous. 00:43:26.080 --> 00:43:29.700 align:middle line:90% 00:43:29.700 --> 00:43:30.200 align:middle line:90% OK. 00:43:30.200 --> 00:43:31.575 align:middle line:84% We're going to use this theorem-- 00:43:31.575 --> 00:43:34.610 align:middle line:90% 00:43:34.610 --> 00:43:36.950 align:middle line:84% I keep saying this although I already erased it, 00:43:36.950 --> 00:43:42.380 align:middle line:84% but the theorem that equates continuity 00:43:42.380 --> 00:43:46.970 align:middle line:84% at a point and the fact that every sequence converges-- 00:43:46.970 --> 00:43:50.990 align:middle line:84% every sequence converging to c implies f of xn converges 00:43:50.990 --> 00:43:51.710 align:middle line:90% to f of c. 00:43:51.710 --> 00:44:03.110 align:middle line:84% We're going to use that theorem to get corresponding theorems 00:44:03.110 --> 00:44:06.500 align:middle line:84% about continuity that look analogous to statements 00:44:06.500 --> 00:44:08.030 align:middle line:90% we made about sequences. 00:44:08.030 --> 00:44:22.780 align:middle line:90% 00:44:22.780 --> 00:44:26.680 align:middle line:84% And then we'll consider a case which is not covered, 00:44:26.680 --> 00:44:30.917 align:middle line:84% which has no analog for sequences. 00:44:30.917 --> 00:44:32.500 align:middle line:84% So let me state the following theorem. 00:44:32.500 --> 00:44:37.555 align:middle line:84% So suppose s is a subset of r, c is an element of s. 00:44:37.555 --> 00:44:40.620 align:middle line:90% 00:44:40.620 --> 00:44:48.320 align:middle line:84% And I have two functions from s to R. 00:44:48.320 --> 00:45:01.180 align:middle line:84% Then if f and g are continuous at c, 00:45:01.180 --> 00:45:08.760 align:middle line:84% then the conclusion is f plus g-- 00:45:08.760 --> 00:45:16.040 align:middle line:84% the function f plus g is continuous at c. 00:45:16.040 --> 00:45:22.770 align:middle line:90% f times g is continuous at c. 00:45:22.770 --> 00:45:34.690 align:middle line:84% And if g of x does not equal 0 for all x and s, 00:45:34.690 --> 00:45:49.640 align:middle line:84% so I can divide by it, then f over g is continuous at c. 00:45:49.640 --> 00:45:56.210 align:middle line:90% 00:45:56.210 --> 00:45:59.570 align:middle line:84% And so for example, I'll prove the first statement, 00:45:59.570 --> 00:46:04.410 align:middle line:84% the other two, you can prove for yourself. 00:46:04.410 --> 00:46:06.900 align:middle line:84% And again, this just follows from this characterization 00:46:06.900 --> 00:46:10.100 align:middle line:84% we have of a function being continuous at a point in terms 00:46:10.100 --> 00:46:11.630 align:middle line:90% of limits of sequences. 00:46:11.630 --> 00:46:14.030 align:middle line:84% And limits of sequences we know well. 00:46:14.030 --> 00:46:16.850 align:middle line:84% We've proven all these properties about them. 00:46:16.850 --> 00:46:19.940 align:middle line:84% If you like, that's where we did all the hard work. 00:46:19.940 --> 00:46:24.860 align:middle line:84% And now getting some payoff in that 00:46:24.860 --> 00:46:29.370 align:middle line:84% we get interesting statements without a whole lot of work. 00:46:29.370 --> 00:46:34.090 align:middle line:84% So we're assuming f and g are continuous at c, 00:46:34.090 --> 00:46:37.720 align:middle line:84% so let's prove f plus g is continuous at c 00:46:37.720 --> 00:46:41.230 align:middle line:84% via this sequential characterization. 00:46:41.230 --> 00:46:44.320 align:middle line:84% Suppose xn is a sequence converging to c. 00:46:44.320 --> 00:46:47.340 align:middle line:90% 00:46:47.340 --> 00:46:58.060 align:middle line:84% Then since f and g are continuous at c, 00:46:58.060 --> 00:47:01.570 align:middle line:84% this implies that the sequences f 00:47:01.570 --> 00:47:08.170 align:middle line:84% of x sub n converges to f of c, and g of x sub n 00:47:08.170 --> 00:47:11.580 align:middle line:90% converges to g of c. 00:47:11.580 --> 00:47:16.440 align:middle line:84% And therefore, by the theorem we proved several lectures ago 00:47:16.440 --> 00:47:18.630 align:middle line:84% that the sum of two convergent sequences 00:47:18.630 --> 00:47:21.600 align:middle line:84% converges to the sum of the limits, 00:47:21.600 --> 00:47:28.140 align:middle line:84% you get that f plus g of x sub n, which is f of x sub n 00:47:28.140 --> 00:47:38.170 align:middle line:84% plus g of x sub n, converges to f of c plus g of c or function 00:47:38.170 --> 00:47:41.250 align:middle line:90% f plus g evaluated at c. 00:47:41.250 --> 00:47:43.680 align:middle line:90% And that's it. 00:47:43.680 --> 00:47:45.660 align:middle line:84% We've now shown that every sequence converging 00:47:45.660 --> 00:47:49.290 align:middle line:84% to c, f plus g of x sub n converges to f plus g of c. 00:47:49.290 --> 00:47:52.120 align:middle line:90% 00:47:52.120 --> 00:47:54.310 align:middle line:84% And similarly for the others, although there's maybe 00:47:54.310 --> 00:47:58.540 align:middle line:84% a little bit of a small hiccup here in that you want to-- 00:47:58.540 --> 00:47:59.320 align:middle line:90% well, no. 00:47:59.320 --> 00:48:04.110 align:middle line:90% 00:48:04.110 --> 00:48:05.565 align:middle line:90% So this follows-- 00:48:05.565 --> 00:48:06.940 align:middle line:84% I was thinking of something else. 00:48:06.940 --> 00:48:09.880 align:middle line:90% Don't worry about that. 00:48:09.880 --> 00:48:12.270 align:middle line:84% But 2 and 3 also follow similarly 00:48:12.270 --> 00:48:13.800 align:middle line:90% using the sequential-- 00:48:13.800 --> 00:48:16.360 align:middle line:90% 00:48:16.360 --> 00:48:18.910 align:middle line:84% the analogous sequential theorems. 00:48:18.910 --> 00:48:39.640 align:middle line:90% 00:48:39.640 --> 00:48:45.940 align:middle line:90% However, we do have-- 00:48:45.940 --> 00:48:48.340 align:middle line:84% so these are three natural operations 00:48:48.340 --> 00:48:49.720 align:middle line:90% we can do with two functions. 00:48:49.720 --> 00:48:50.495 align:middle line:90% We can add them. 00:48:50.495 --> 00:48:51.370 align:middle line:90% We can multiply them. 00:48:51.370 --> 00:48:54.400 align:middle line:84% We can divide them, just like we had for sequences. 00:48:54.400 --> 00:48:56.920 align:middle line:84% There is one operation we can do with functions 00:48:56.920 --> 00:48:59.560 align:middle line:84% that you don't do with sequences or can't do with sequences, 00:48:59.560 --> 00:49:02.080 align:middle line:90% and that's compose them. 00:49:02.080 --> 00:49:04.930 align:middle line:84% So the natural question is, is the composition 00:49:04.930 --> 00:49:07.510 align:middle line:84% of two continuous functions continuous? 00:49:07.510 --> 00:49:10.510 align:middle line:90% And the answer to that is yes. 00:49:10.510 --> 00:49:12.440 align:middle line:90% We have to state this carefully. 00:49:12.440 --> 00:49:27.220 align:middle line:84% So suppose a and b are a subset of R, and c is an element of a. 00:49:27.220 --> 00:49:31.410 align:middle line:90% 00:49:31.410 --> 00:49:36.770 align:middle line:84% So let's write this a little bit differently. 00:49:36.770 --> 00:49:41.793 align:middle line:84% Let a and b be a subset of r. c is an element of a, 00:49:41.793 --> 00:49:45.120 align:middle line:90% and f will be a function from-- 00:49:45.120 --> 00:49:47.613 align:middle line:90% 00:49:47.613 --> 00:49:48.530 align:middle line:90% let me get this right. 00:49:48.530 --> 00:49:52.960 align:middle line:90% 00:49:52.960 --> 00:49:57.520 align:middle line:84% f will be a function from a to r, 00:49:57.520 --> 00:50:02.210 align:middle line:84% and g is a function from a set b to a. 00:50:02.210 --> 00:50:02.710 align:middle line:90% OK. 00:50:02.710 --> 00:50:03.960 align:middle line:90% So I'm getting this backwards. 00:50:03.960 --> 00:50:09.170 align:middle line:90% 00:50:09.170 --> 00:50:12.470 align:middle line:84% So when I compose these two, when I take f of g, f of g 00:50:12.470 --> 00:50:16.910 align:middle line:84% will be a function now from b to r. 00:50:16.910 --> 00:50:33.730 align:middle line:84% So if g is continuous at c and f is continuous at the point 00:50:33.730 --> 00:50:36.040 align:middle line:84% g of c, which is, remember, an element 00:50:36.040 --> 00:50:50.610 align:middle line:84% of a, then the composition f of g is continuous at c. 00:50:50.610 --> 00:50:56.850 align:middle line:90% 00:50:56.850 --> 00:50:59.490 align:middle line:84% So we'll use, again, this characterization 00:50:59.490 --> 00:51:03.040 align:middle line:90% in terms of sequences. 00:51:03.040 --> 00:51:04.930 align:middle line:90% We don't necessarily have to. 00:51:04.930 --> 00:51:09.600 align:middle line:84% We could have done it, strictly speaking, from the definition. 00:51:09.600 --> 00:51:13.870 align:middle line:84% But this is a nice short way of proving this statement. 00:51:13.870 --> 00:51:25.540 align:middle line:84% So let xn be a sequence in b, so of elements of b such 00:51:25.540 --> 00:51:28.270 align:middle line:90% that x sub n converges to c. 00:51:28.270 --> 00:51:34.950 align:middle line:84% And what we want to show is that f of g limit 00:51:34.950 --> 00:51:43.840 align:middle line:84% as n goes to infinity of f of g of x sub n equals f of g of c. 00:51:43.840 --> 00:51:46.990 align:middle line:90% 00:51:46.990 --> 00:51:50.380 align:middle line:90% Let me put this off to the side. 00:51:50.380 --> 00:51:50.880 align:middle line:90% OK. 00:51:50.880 --> 00:52:09.200 align:middle line:84% So since xn converges to c and g is continuous at c, 00:52:09.200 --> 00:52:18.320 align:middle line:84% this implies that the sequence g of xn converges to g of c. 00:52:18.320 --> 00:52:20.120 align:middle line:90% Now, g of x sub n-- 00:52:20.120 --> 00:52:25.970 align:middle line:84% this is a sequence now in a converging to an element of a. 00:52:25.970 --> 00:52:27.140 align:middle line:90% Yeah? 00:52:27.140 --> 00:52:35.210 align:middle line:84% So since this sequence g of x sub n converges to g of c 00:52:35.210 --> 00:52:44.570 align:middle line:84% and g, and now f, is continuous at g of c, 00:52:44.570 --> 00:52:54.260 align:middle line:84% this implies that f of g of x sub n converges to f 00:52:54.260 --> 00:53:02.060 align:middle line:84% of g of c, i.e., this is, by definition, the same as saying 00:53:02.060 --> 00:53:10.610 align:middle line:84% f of g of x sub n equals f of g of c. 00:53:10.610 --> 00:53:16.270 align:middle line:90% 00:53:16.270 --> 00:53:20.140 align:middle line:84% So we have an operation that we can't do with sequences, 00:53:20.140 --> 00:53:22.780 align:middle line:84% but this operation still preserves continuity. 00:53:22.780 --> 00:53:29.080 align:middle line:84% So taking sums of continuous functions 00:53:29.080 --> 00:53:30.558 align:middle line:90% maintains a continuous-- 00:53:30.558 --> 00:53:32.350 align:middle line:84% stays in the class of continuous functions. 00:53:32.350 --> 00:53:34.810 align:middle line:84% The product of two continuous functions is continuous. 00:53:34.810 --> 00:53:39.590 align:middle line:90% The quotient is continuous. 00:53:39.590 --> 00:53:45.200 align:middle line:84% And now also the composition of two continuous functions 00:53:45.200 --> 00:53:46.280 align:middle line:90% is continuous. 00:53:46.280 --> 00:53:51.350 align:middle line:90% 00:53:51.350 --> 00:53:56.750 align:middle line:84% So as a consequence of this, we can use this to prove some-- 00:53:56.750 --> 00:54:00.860 align:middle line:90% 00:54:00.860 --> 00:54:03.540 align:middle line:84% give more examples of functions which are continuous. 00:54:03.540 --> 00:54:12.630 align:middle line:84% So for example, for all n, a natural number, 00:54:12.630 --> 00:54:22.240 align:middle line:84% f of x equals x to the n is continuous as a function 00:54:22.240 --> 00:54:24.595 align:middle line:90% for x and r. 00:54:24.595 --> 00:54:25.625 align:middle line:90% So what's the proof? 00:54:25.625 --> 00:54:28.140 align:middle line:90% 00:54:28.140 --> 00:54:30.270 align:middle line:90% We can do this by induction. 00:54:30.270 --> 00:54:43.355 align:middle line:84% So for n equals 1, f of x equals x, we've already done. 00:54:43.355 --> 00:54:47.083 align:middle line:90% 00:54:47.083 --> 00:54:48.500 align:middle line:84% That was one of the first examples 00:54:48.500 --> 00:54:51.230 align:middle line:84% of continuous functions we did, was ax plus b. 00:54:51.230 --> 00:54:56.840 align:middle line:84% So a equals 1, b equals 0 gives me this case. 00:54:56.840 --> 00:54:58.100 align:middle line:90% So this is the base case. 00:54:58.100 --> 00:55:01.220 align:middle line:90% Let's now do the inductive step. 00:55:01.220 --> 00:55:06.110 align:middle line:90% 00:55:06.110 --> 00:55:12.230 align:middle line:84% Suppose that the function x to the m is continuous. 00:55:12.230 --> 00:55:15.950 align:middle line:84% And now we want to show x to the n plus 1 is continuous. 00:55:15.950 --> 00:55:18.770 align:middle line:90% 00:55:18.770 --> 00:55:26.860 align:middle line:84% Then x to the m plus 1, this is equal to x times x to the m. 00:55:26.860 --> 00:55:29.890 align:middle line:84% And since x to the m is continuous 00:55:29.890 --> 00:55:33.370 align:middle line:84% and the function f of x equals x is continuous, 00:55:33.370 --> 00:55:36.070 align:middle line:84% the product of two continuous functions is continuous. 00:55:36.070 --> 00:55:47.290 align:middle line:90% 00:55:47.290 --> 00:56:01.460 align:middle line:84% It's a product of two continuous functions, which by the theorem 00:56:01.460 --> 00:56:10.530 align:middle line:84% we proved implies that x to the n plus 1 is continuous. 00:56:10.530 --> 00:56:14.170 align:middle line:84% So I keep saying continuous, but what I'm saying is for all c 00:56:14.170 --> 00:56:18.180 align:middle line:84% and r, x to the m is continuous at c. 00:56:18.180 --> 00:56:20.472 align:middle line:90% 00:56:20.472 --> 00:56:22.180 align:middle line:84% So maybe I should have written that down, 00:56:22.180 --> 00:56:26.380 align:middle line:84% but I think that meaning should be clear enough. 00:56:26.380 --> 00:56:33.690 align:middle line:84% And by the same inductive reasoning, 00:56:33.690 --> 00:56:43.040 align:middle line:84% you could also prove that in a natural number, polynomials 00:56:43.040 --> 00:56:46.370 align:middle line:90% are continuous. f of x-- 00:56:46.370 --> 00:56:57.105 align:middle line:84% for all in a natural number, a0 and R, 00:56:57.105 --> 00:57:02.890 align:middle line:84% the function given by a polynomial a sub n x to the n 00:57:02.890 --> 00:57:11.180 align:middle line:84% plus a sub n minus 1, x to the n minus 1 plus a sub 0 00:57:11.180 --> 00:57:12.080 align:middle line:90% is continuous. 00:57:12.080 --> 00:57:18.250 align:middle line:90% 00:57:18.250 --> 00:57:21.520 align:middle line:84% So rather than write down the actual proof by induction, 00:57:21.520 --> 00:57:23.960 align:middle line:90% let's talk our way through it. 00:57:23.960 --> 00:57:26.530 align:middle line:84% So again, for n equals 1, this is just 00:57:26.530 --> 00:57:29.620 align:middle line:84% going to be some number times x, which we've already dealt 00:57:29.620 --> 00:57:31.550 align:middle line:90% with in a previous example. 00:57:31.550 --> 00:57:32.960 align:middle line:90% So we know that's continuous. 00:57:32.960 --> 00:57:35.020 align:middle line:90% So that settles the base case. 00:57:35.020 --> 00:57:38.530 align:middle line:84% Now let's assume that we've proved this for n 00:57:38.530 --> 00:57:42.310 align:middle line:84% equals m, or let's assume this for n equals m 00:57:42.310 --> 00:57:45.820 align:middle line:84% and look at a function with n equals n plus 1. 00:57:45.820 --> 00:57:50.410 align:middle line:84% That's going to be a to the a sub m plus 1 times x 00:57:50.410 --> 00:57:53.740 align:middle line:84% to the n plus 1 plus a lower-order polynomial, which 00:57:53.740 --> 00:57:56.020 align:middle line:90% we already know is continuous. 00:57:56.020 --> 00:58:00.150 align:middle line:84% And a sub m plus 1 times x to the n 00:58:00.150 --> 00:58:02.830 align:middle line:84% plus 1, that's continuous because it's 00:58:02.830 --> 00:58:06.190 align:middle line:84% the product of a constant and a continuous function 00:58:06.190 --> 00:58:07.910 align:middle line:90% by the previous example. 00:58:07.910 --> 00:58:10.900 align:middle line:84% So that would be continuous plus the lower order 00:58:10.900 --> 00:58:13.630 align:middle line:84% polynomial, which is continuous by assumption. 00:58:13.630 --> 00:58:16.540 align:middle line:84% That will be continuous by the theorem that we proved. 00:58:16.540 --> 00:58:21.100 align:middle line:90% 00:58:21.100 --> 00:58:25.220 align:middle line:84% But you don't have to just look at polynomials. 00:58:25.220 --> 00:58:34.510 align:middle line:84% For example-- so for example, the function f of x 00:58:34.510 --> 00:58:41.020 align:middle line:84% equals 1 over 3 plus sine x to the 4th. 00:58:41.020 --> 00:58:43.060 align:middle line:90% This is also continuous-- 00:58:43.060 --> 00:58:47.230 align:middle line:90% 00:58:47.230 --> 00:58:50.750 align:middle line:90% is a continuous function. 00:58:50.750 --> 00:58:51.250 align:middle line:90% Why? 00:58:51.250 --> 00:58:54.430 align:middle line:84% Because for all x-- first off, 3 plus sine x to the 4, 00:58:54.430 --> 00:58:56.140 align:middle line:90% this is never 0. 00:58:56.140 --> 00:58:58.820 align:middle line:84% So this function makes perfectly good sense. 00:58:58.820 --> 00:59:02.440 align:middle line:84% So the bottom is always non-zero. 00:59:02.440 --> 00:59:06.790 align:middle line:84% Sine x is continuous, so sine x to the 4 is continuous. 00:59:06.790 --> 00:59:08.475 align:middle line:90% 3 is a continuous function. 00:59:08.475 --> 00:59:09.350 align:middle line:90% It's just a constant. 00:59:09.350 --> 00:59:15.640 align:middle line:84% So the bottom as a function on its own is continuous. 00:59:15.640 --> 00:59:18.370 align:middle line:84% And 1 over that function is continuous 00:59:18.370 --> 00:59:22.570 align:middle line:84% as long as the bottom is never 0, which it's never 0, 00:59:22.570 --> 00:59:26.270 align:middle line:90% so again, by this theorem here. 00:59:26.270 --> 00:59:29.500 align:middle line:84% So we used the composition one to say that sine x to the 4 00:59:29.500 --> 00:59:31.150 align:middle line:90% is continuous. 00:59:31.150 --> 00:59:34.120 align:middle line:84% And we use that one, the theorem before that, 00:59:34.120 --> 00:59:37.390 align:middle line:84% to show that 1 over 3 plus sine x 4 is continuous. 00:59:37.390 --> 00:59:39.310 align:middle line:84% So let me write this out real quick. 00:59:39.310 --> 00:59:44.530 align:middle line:90% 00:59:44.530 --> 00:59:47.680 align:middle line:84% I will say by composition theorem, 00:59:47.680 --> 00:59:56.060 align:middle line:84% the function sine x of 4 is continuous, 00:59:56.060 --> 01:00:00.310 align:middle line:84% which implies by, I'll say, algebraic theorem, 01:00:00.310 --> 01:00:07.350 align:middle line:84% meaning that theorem that involves algebraic operations, 01:00:07.350 --> 01:00:15.580 align:middle line:84% 3 plus sine x to the 4 is continuous. 01:00:15.580 --> 01:00:22.260 align:middle line:84% And by the algebraic theorem again, 01:00:22.260 --> 01:00:32.520 align:middle line:84% since 3 plus sine x to the 4 is never 0 for all x and r, 01:00:32.520 --> 01:00:33.540 align:middle line:90% this is continuous. 01:00:33.540 --> 01:00:37.640 align:middle line:90% 01:00:37.640 --> 01:00:39.110 align:middle line:90% All right. 01:00:39.110 --> 01:00:41.390 align:middle line:84% So next time, we'll look at some properties 01:00:41.390 --> 01:00:53.450 align:middle line:84% of continuous functions, namely called the min and max theorems 01:00:53.450 --> 01:00:54.515 align:middle line:90% for continuous functions. 01:00:54.515 --> 01:00:56.390 align:middle line:84% So the great thing about continuous functions 01:00:56.390 --> 01:00:59.090 align:middle line:84% is if you look at them on closed and bounded intervals, 01:00:59.090 --> 01:01:02.720 align:middle line:84% they always attain a maximum and minimum at some point-- 01:01:02.720 --> 01:01:09.660 align:middle line:84% not just that the graph is bounded above or below, 01:01:09.660 --> 01:01:12.840 align:middle line:84% but there's actual point where f reaches that min 01:01:12.840 --> 01:01:14.010 align:middle line:90% and reaches that max. 01:01:14.010 --> 01:01:17.460 align:middle line:84% And then we'll also prove the intermediate value theorem, 01:01:17.460 --> 01:01:18.750 align:middle line:90% which says between any two-- 01:01:18.750 --> 01:01:22.090 align:middle line:90% 01:01:22.090 --> 01:01:26.470 align:middle line:84% if I have a continuous function on an interval a, b, 01:01:26.470 --> 01:01:29.590 align:middle line:84% and I look at f of a and f of b, and I take a number 01:01:29.590 --> 01:01:32.620 align:middle line:84% in between those two values, f of a and f of b, 01:01:32.620 --> 01:01:34.490 align:middle line:84% there exists a point in between them 01:01:34.490 --> 01:01:38.740 align:middle line:90% so that f attains that value. 01:01:38.740 --> 01:01:42.240 align:middle line:90% And this is extremely important. 01:01:42.240 --> 01:01:45.090 align:middle line:90% 01:01:45.090 --> 01:01:47.700 align:middle line:84% And we'll definitely see why later once we 01:01:47.700 --> 01:01:52.310 align:middle line:84% get to differentiability and continuity. 01:01:52.310 --> 01:01:58.000 align:middle line:90%