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Sal introduces a formal definition of continuity at a point using limits. Created by Sal Khan.
Video transcript
What I want to do in this video is talk about continuity. Continuity of a function is something that is pretty easy to recognize when you see it. But we'll also talk about how we can more rigorously define it. So when I talk about it being "pretty easy to identify" is Let me draw the y-axis, that is the x-axis. ...and if I were to draw a function, let's say f(x) looks something like this. And I would say over the interval that I've drawn it... so it looks like from x=0 to that point right over there. Is this function continuous? Well, you would say "No it isn't". Look, over here we see the function just jumps all of a sudden. >From this point to this point right over here. This is NOT continuous. You might even say we have a discontinuity at this value of x right over here. We would call this a discontinuity. And this type of discontinuity is called a "Jump" discontinuity. So you would say this is not continuous. It is obvious these two things do not connect. They don't touch each other. Similarly, if you were to look at a function that looked like - let me draw another one y and x. And let's say the function looks something like this... maybe right over here, it looks like this. And then the function is defined to be this point. Right over there. Is the function continuous over the interval that I've depicted right over here. And you would immediately say "no it isn't" because right over at this point the function goes up to this point. Just like this. And this kind of discontinuity called a "Removeable" discontinuity. One could make a reasonable argument that this also looks like a Jump. But this is typically categorized as Removeable. ...because if you just redefine the function so it wasn't up here... ...but it was right over here, then the function is continuous. So you can kind of... ...remove the discontinuity. And finally if I were to draw another function. So let me draw another one right over here. X, Y. And ask you "is this one continuous over the interval I've depicted". And you'd say "Well, yeah, it looks connected all the way. There aren't any jumps over here... ...or Reomveable discontinuities over here. This one looks continuous." Continuous. And you'd be right. So that's the general sense of continuity. And you can kind of spot it when you see it. But let's think about a more rigourous definition of one. And since we already have a definition of limits... epsilon-delta definition, gives us a rigorous definition for limits. So we can prove when a limit exists and what the value of that limit is. Let's use that to create a rigorous definition of continuity. So let's think about a function over some type of an interval. So let's say that we have... so let me draw another function. Let me draw some type of a function. And then we'll see whether our more rigorous definition of continuity passes muster when we look at all of these things up here. Let me draw an interval. Right over here. So it's between this x-value and that x-value. This is the x-axis and this is the y-axis. And let me draw my function over that interval. It looks something like this. So we say that a function is continuous at an interior point. So an "interior point" is a point that is not at the edge of my boundary. So this is an interior point for my interval. This would be an endpoint and this would also be an endpoint. We say it's continuous at an interior point interior to my interval, means that the limit at interior point c. So this is the point x=c. We can say it is continuous at the interior point c if the limit of our... function - this is our function right over here - as x... approaches c is equal to the value of our function. Now, does this make sense? Well, what we're saying is that that point - well this is f(c) right over there - and the limit as we approach that is the same thing as the value of the function. Which makes a lot of sense. Now let's think about it if these would have somehow been able to pass for continuous, in that context. Well, over here, let's say that this is our point c. f(c) is right over there. That is f(c). Now is it the case that the limit of f(x), as x approaches c, is equal to f(c). Well, if we take the limit of f(x), as x approaches c from the positive direction it does look like it is f(c). But if we take the limit, this does not equal, does NOT equal, the limit of f(x) as x approaches c from the negative direction. As we go from the negative direction, we're not approaching f(c). So therefore, this does not hold up. In order for the limit to be equal to f(c), the limit from both directions needs to be equal to it. And this is not the case. So this would not pass muster by our formal definition. Which is good, because we see visually that this one is not continuous. What about this one right over here. And let me re-set it up. Let me make sure that looks like a hole over there. So we see here... what is the limit - and this is our c, right over here - the limit... of f(x) as x approaches c. Let's say that is equal to L. So that, we've seen many limits like this before. That's L right over there. And it's pretty clear just looking at this that L does not equal f(c). This right over here. f(c). So once again, this would not pass our test. The limit of f(x), as x approaches c, which is this right over here, is not equal to f(c). So this would not pass our test. And here, any of the interior points would pass our test. The limit, as x approaches this value, is indeed equal to the function evaluated at that point. So it seems to be good for all of those. Now let's give a definition for when we're talking about boundary points. So this is continuity for an interior point. And let's think about continuity - I'll do it right over here - at end point c. So let's first consider the left endpoint. If left endpoint - so what am I talking about, a "left endpoint"? Let me draw my axes. X-axis. Y-axis. And let me draw my interval. So this is the left endpoint of my interval. This is the right end point of my interval. And let me draw my function over that interval. Looks something like this. So we're talking about a left endpoint, we're talking about our c being right over here. It is the left endpoint. So if we're talking about a left endpoint, we are continuous at c means... or to say that we're continuous at this left endpoing c, that means... that the limit of f(x) as x approaches c - well we can't even approach c from the left hand side, we have to approach from the right. is equal to f(c). And so this is really kind of a - we can only approach from one direction. So we can't really say the limit in general, but we can take the limit from one side. So it's very similar to what we just said for an interior point. And we see over here, it is indeed the case as x approaches c ...our function is approaching this point over here ...which is the exact same thing as f(c). So we are continuous at that point. What's an example of an endpoint where we would not be continuous at an endpoint? Well, I can imagine a graph that looks something like this. So here's our interval. And maybe our function. So at c it looks like that. There's a little hole right here. Or there's no hole, the function just has a removeable discontinuity over there. Or at least visually it looks like that. And you can see visually, this would not pass the test. Because the limit, as we approach c from the positive direction. is right over here. That's the limit. But f(c) is up here. So f(c) does not equal the limit as x approaches c from the positive direction. So this would not be continuous. And you could imagine what we do if we're dealing with the right endpoing. So, we say we're continuous at right endpoint c if so let me draw that. I'll do my best attempt to draw it. So this is my x-axis. This is my y-axis. Let me draw my interval. So that I care about. A right endpoint means that c is right over there. And we can say that we're continuous at... the function is continuous at... x equals c means that the limit of f(x)... as x approaches c - now we can't approach it from both sides. We can only approach it from the left hand side. As x approaches c from the negative direction. We could say that, if this is true, then this implies that we're continuous at that right endpoing c. And vice versa. And a situation where we're not? Well, we could imagine this being defined right at that point you could say the function jumps up. Jus tlike we did right over there. So once again, continuity - not a really hard to fathom idea. Whenever you see the function just all of a sudden jumping. Or there is a gap in it, it's a pretty good sense that the function is not connected there. It's not continuous. But what we did in this video is we used limits to define a more rigorous definition of continuity.