1 00:00:00,837 --> 00:00:05,236 What I want to do in this video is talk about continuity. 2 00:00:05,236 --> 00:00:10,302 Continuity of a function is something that is pretty easy to recognize when you see it. 3 00:00:10,302 --> 00:00:15,126 But we'll also talk about how we can more rigorously define it. 4 00:00:15,126 --> 00:00:18,508 So when I talk about it being "pretty easy to identify" is 5 00:00:18,508 --> 00:00:24,302 Let me draw the y-axis, that is the x-axis. 6 00:00:24,302 --> 00:00:34,371 ...and if I were to draw a function, let's say f(x) looks something like this. 7 00:00:34,371 --> 00:00:36,717 And I would say over the interval that I've drawn it... 8 00:00:36,717 --> 00:00:40,568 so it looks like from x=0 to that point right over there. 9 00:00:40,568 --> 00:00:44,233 Is this function continuous? Well, you would say "No it isn't". 10 00:00:44,233 --> 00:00:47,626 Look, over here we see the function just jumps all of a sudden. 11 00:00:47,626 --> 00:00:49,969 From this point to this point right over here. 12 00:00:49,969 --> 00:00:54,199 This is NOT continuous. 13 00:00:54,199 --> 00:01:00,969 You might even say we have a discontinuity at this value of x right over here. 14 00:01:00,969 --> 00:01:02,956 We would call this a discontinuity. 15 00:01:02,956 --> 00:01:08,696 And this type of discontinuity is called a "Jump" discontinuity. 16 00:01:08,696 --> 00:01:13,302 So you would say this is not continuous. It is obvious these two things do not connect. 17 00:01:13,302 --> 00:01:15,032 They don't touch each other. 18 00:01:15,032 --> 00:01:19,289 Similarly, if you were to look at a function that looked like - let me draw another one 19 00:01:19,289 --> 00:01:26,207 y and x. And let's say the function looks something like this... maybe 20 00:01:26,207 --> 00:01:31,235 right over here, it looks like this. And then the function is defined to be this point. 21 00:01:31,235 --> 00:01:32,454 Right over there. 22 00:01:32,454 --> 00:01:37,202 Is the function continuous over the interval that I've depicted right over here. 23 00:01:37,202 --> 00:01:43,237 And you would immediately say "no it isn't" because right over at this point the function goes up to this point. 24 00:01:43,237 --> 00:01:48,673 Just like this. And this kind of discontinuity 25 00:01:48,673 --> 00:01:54,109 ...is called a "Removeable" discontinuity. 26 00:01:54,109 --> 00:02:02,037 One could make a reasonable argument that this also looks like a Jump. But this is typically categorized as Removeable. 27 00:02:02,037 --> 00:02:06,302 ...because if you just redefine the function so it wasn't up here... 28 00:02:06,302 --> 00:02:10,641 ...but it was right over here, then the function is continuous. So you can kind of... 29 00:02:10,641 --> 00:02:14,169 ...remove the discontinuity. 30 00:02:14,169 --> 00:02:17,651 And finally if I were to draw another function. 31 00:02:17,682 --> 00:02:22,270 So let me draw another one right over here. X, Y. 32 00:02:22,301 --> 00:02:27,036 And ask you "is this one continuous over the interval I've depicted". 33 00:02:27,036 --> 00:02:31,570 And you'd say "Well, yeah, it looks connected all the way. There aren't any jumps over here... 34 00:02:31,570 --> 00:02:35,302 ...or Reomveable discontinuities over here. This one looks continuous." 35 00:02:35,302 --> 00:02:38,585 Continuous. And you'd be right. 36 00:02:38,585 --> 00:02:40,759 So that's the general sense of continuity. 37 00:02:40,789 --> 00:02:43,043 And you can kind of spot it when you see it. 38 00:02:43,043 --> 00:02:46,302 But let's think about a more rigourous definition of one. 39 00:02:46,302 --> 00:02:49,302 And since we already have a definition of limits... 40 00:02:49,302 --> 00:02:59,908 epsilon-delta definition, gives us a rigorous definition for limits. 41 00:02:59,908 --> 00:03:03,871 So we can prove when a limit exists and what the value of that limit is. 42 00:03:03,871 --> 00:03:08,959 Let's use that to create a rigorous definition of continuity. 43 00:03:08,959 --> 00:03:14,903 So let's think about a function over some type of an interval. 44 00:03:14,903 --> 00:03:18,907 So let's say that we have... so let me draw another function. 45 00:03:18,907 --> 00:03:21,787 Let me draw some type of a function. 46 00:03:21,787 --> 00:03:29,644 And then we'll see whether our more rigorous definition of continuity passes muster when we look at all of these things up here. 47 00:03:29,644 --> 00:03:32,635 Let me draw an interval. Right over here. 48 00:03:32,635 --> 00:03:37,636 So it's between this x-value and that x-value. This is the x-axis and this is the y-axis. 49 00:03:37,636 --> 00:03:40,705 And let me draw my function over that interval. 50 00:03:40,705 --> 00:03:44,303 It looks something like this. 51 00:03:44,303 --> 00:03:49,117 So we say that a function is continuous at an interior point. 52 00:03:49,117 --> 00:03:55,367 So an "interior point" is a point that is not at the edge of my boundary. 53 00:03:55,367 --> 00:03:58,041 So this is an interior point for my interval. 54 00:03:58,041 --> 00:04:00,169 This would be an endpoint and this would also be an endpoint. 55 00:04:00,169 --> 00:04:07,666 We say it's continuous at an interior point 56 00:04:07,666 --> 00:04:10,085 interior to my interval, means 57 00:04:10,085 --> 00:04:16,704 that the limit at interior point c. 58 00:04:16,704 --> 00:04:19,243 So this is the point x=c. 59 00:04:19,243 --> 00:04:25,619 We can say it is continuous at the interior point c if the limit of our... 60 00:04:25,619 --> 00:04:29,619 function - this is our function right over here - as x... 61 00:04:29,619 --> 00:04:34,900 approaches c is equal to the value of our function. 62 00:04:34,900 --> 00:04:36,453 Now, does this make sense? 63 00:04:36,453 --> 00:04:40,596 Well, what we're saying is that that point - well this is f(c) right over there - 64 00:04:40,627 --> 00:04:44,998 and the limit as we approach that is the same thing as the value of the function. 65 00:04:44,998 --> 00:04:46,502 Which makes a lot of sense. 66 00:04:46,502 --> 00:04:53,042 Now let's think about it if these would have somehow been able to pass for continuous, in that context. 67 00:04:53,042 --> 00:04:57,701 Well, over here, let's say that this is our point c. 68 00:04:57,701 --> 00:05:00,283 f(c) is right over there. 69 00:05:00,283 --> 00:05:04,977 That is f(c). Now is it the case that the limit 70 00:05:04,977 --> 00:05:10,201 of f(x), as x approaches c, is equal to f(c). 71 00:05:10,216 --> 00:05:16,906 Well, if we take the limit of f(x), as x approaches c from the positive direction 72 00:05:16,906 --> 00:05:22,789 it does look like it is f(c). 73 00:05:22,789 --> 00:05:28,637 But if we take the limit, this does not equal, does NOT equal, the limit 74 00:05:28,637 --> 00:05:35,102 of f(x) as x approaches c from the negative direction. 75 00:05:35,102 --> 00:05:37,842 As we go from the negative direction, we're not approaching f(c). 76 00:05:37,842 --> 00:05:40,975 So therefore, this does not hold up. 77 00:05:40,975 --> 00:05:44,125 In order for the limit to be equal to f(c), the limit from both 78 00:05:44,125 --> 00:05:47,242 directions needs to be equal to it. And this is not the case. 79 00:05:47,242 --> 00:05:50,969 So this would not pass muster by our formal definition. 80 00:05:50,969 --> 00:05:54,968 Which is good, because we see visually that this one is not continuous. 81 00:05:54,968 --> 00:05:57,087 What about this one right over here. 82 00:05:57,102 --> 00:06:01,970 And let me re-set it up. Let me make sure that looks like a hole over there. 83 00:06:01,970 --> 00:06:09,128 So we see here... what is the limit - and this is our c, right over here - the limit... 84 00:06:09,128 --> 00:06:11,369 of f(x) as x approaches c. 85 00:06:11,369 --> 00:06:13,297 Let's say that is equal to L. 86 00:06:13,297 --> 00:06:17,033 So that, we've seen many limits like this before. 87 00:06:17,033 --> 00:06:18,302 That's L right over there. 88 00:06:18,302 --> 00:06:23,909 And it's pretty clear just looking at this that L does not equal f(c). 89 00:06:23,909 --> 00:06:26,450 This right over here. 90 00:06:26,450 --> 00:06:27,502 ...is f(c). 91 00:06:27,502 --> 00:06:29,541 So once again, this would not pass our test. 92 00:06:29,541 --> 00:06:33,209 The limit of f(x), as x approaches c, which is this right over here, 93 00:06:33,209 --> 00:06:38,164 is not equal to f(c). So this would not pass our test. 94 00:06:38,164 --> 00:06:41,037 And here, any of the interior points would pass our test. 95 00:06:41,037 --> 00:06:52,450 The limit, as x approaches this value, is indeed equal to the function evaluated at that point. 96 00:06:52,450 --> 00:06:54,457 So it seems to be good for all of those. 97 00:06:54,457 --> 00:06:57,902 Now let's give a definition for when we're talking about boundary points. 98 00:06:57,902 --> 00:07:00,568 So this is continuity for an interior point. 99 00:07:00,568 --> 00:07:19,261 And let's think about continuity - I'll do it right over here - at end point c. 100 00:07:19,261 --> 00:07:24,829 So let's first consider the left endpoint. 101 00:07:24,829 --> 00:07:31,782 If left endpoint - so what am I talking about, a "left endpoint"? 102 00:07:31,782 --> 00:07:39,368 Let me draw my axes. X-axis. Y-axis. 103 00:07:39,368 --> 00:07:42,125 And let me draw my interval. 104 00:07:42,125 --> 00:07:47,569 So this is the left endpoint of my interval. This is the right end point of my interval. 105 00:07:47,569 --> 00:07:50,954 And let me draw my function over that interval. 106 00:07:50,954 --> 00:07:54,302 Looks something like this. 107 00:07:54,302 --> 00:08:00,569 So we're talking about a left endpoint, we're talking about our c being right over here. 108 00:08:00,569 --> 00:08:02,707 It is the left endpoint. 109 00:08:02,707 --> 00:08:09,455 So if we're talking about a left endpoint, we are continuous at c means... 110 00:08:09,455 --> 00:08:14,637 or to say that we're continuous at this left endpoing c, that means... 111 00:08:14,637 --> 00:08:20,533 that the limit of f(x) as x approaches c - 112 00:08:20,533 --> 00:08:26,794 well we can't even approach c from the left hand side, we have to approach from the right. 113 00:08:26,794 --> 00:08:29,599 is equal to f(c). 114 00:08:29,614 --> 00:08:33,233 And so this is really kind of a - we can only approach from one direction. 115 00:08:33,233 --> 00:08:36,169 So we can't really say the limit in general, but we can take the limit from one side. 116 00:08:36,169 --> 00:08:39,122 So it's very similar to what we just said for an interior point. 117 00:08:39,122 --> 00:08:42,835 And we see over here, it is indeed the case as x approaches c 118 00:08:42,835 --> 00:08:46,302 ...our function is approaching this point over here 119 00:08:46,302 --> 00:08:48,873 ...which is the exact same thing as f(c). 120 00:08:48,873 --> 00:08:50,043 So we are continuous at that point. 121 00:08:50,043 --> 00:08:55,118 What's an example of an endpoint where we would not be continuous at an endpoint? 122 00:08:55,118 --> 00:08:59,836 Well, I can imagine a graph that looks something like this. 123 00:08:59,836 --> 00:09:05,046 So here's our interval. 124 00:09:05,123 --> 00:09:10,902 And maybe our function. So at c it looks like that. There's a little hole right here. 125 00:09:10,902 --> 00:09:15,903 Or there's no hole, the function just has a removeable discontinuity over there. 126 00:09:15,962 --> 00:09:17,456 Or at least visually it looks like that. 127 00:09:17,487 --> 00:09:20,981 And you can see visually, this would not pass the test. 128 00:09:20,981 --> 00:09:23,676 Because the limit, as we approach c from the positive direction. 129 00:09:23,676 --> 00:09:25,063 is right over here. 130 00:09:25,079 --> 00:09:26,471 That's the limit. 131 00:09:26,471 --> 00:09:28,711 But f(c) is up here. 132 00:09:28,711 --> 00:09:33,169 So f(c) does not equal the limit as x approaches c from the positive direction. 133 00:09:33,185 --> 00:09:38,459 So this would not be continuous. 134 00:09:38,459 --> 00:09:41,284 And you could imagine what we do if we're dealing with the right endpoing. 135 00:09:41,284 --> 00:09:51,522 So, we say we're continuous at right endpoint c if 136 00:09:51,522 --> 00:09:55,959 so let me draw that. 137 00:09:55,959 --> 00:09:58,789 I'll do my best attempt to draw it. 138 00:09:58,789 --> 00:10:01,716 So this is my x-axis. This is my y-axis. 139 00:10:01,716 --> 00:10:04,303 Let me draw my interval. 140 00:10:04,303 --> 00:10:06,908 So that I care about. 141 00:10:06,908 --> 00:10:12,569 A right endpoint means that c is right over there. 142 00:10:12,569 --> 00:10:18,707 And we can say that we're continuous at... the function is continuous at... 143 00:10:18,707 --> 00:10:23,908 x equals c means that the limit of f(x)... 144 00:10:23,908 --> 00:10:27,038 as x approaches c - now we can't approach it from both sides. 145 00:10:27,038 --> 00:10:31,302 We can only approach it from the left hand side. 146 00:10:31,302 --> 00:10:35,100 As x approaches c from the negative direction. 147 00:10:35,100 --> 00:10:38,540 We could say that, if this is true, then this implies 148 00:10:38,540 --> 00:10:41,110 that we're continuous at that right endpoing c. 149 00:10:41,110 --> 00:10:43,036 And vice versa. 150 00:10:43,036 --> 00:10:45,503 And a situation where we're not? 151 00:10:45,503 --> 00:10:46,836 Well, we could imagine this being defined right at that point 152 00:10:46,836 --> 00:10:50,456 you could say the function jumps up. 153 00:10:50,456 --> 00:10:52,507 Jus tlike we did right over there. 154 00:10:52,507 --> 00:10:56,700 So once again, continuity - not a really hard to fathom idea. 155 00:10:56,700 --> 00:10:59,640 Whenever you see the function just all of a sudden jumping. 156 00:10:59,640 --> 00:11:03,038 Or there is a gap in it, it's a pretty good sense that 157 00:11:03,038 --> 00:11:05,034 the function is not connected there. 158 00:11:05,034 --> 00:11:06,374 It's not continuous. 159 00:11:06,374 --> 00:11:09,623 But what we did in this video is we used limits 160 00:11:09,623 --> 00:11:13,623 to define a more rigorous definition of continuity.