Continuity at a point
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What I want to do in this video is talk about continuity. And 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 recognize, let me draw some functions here. So let's say this is the y-axis, that is the x-axis. And if I were to draw a function, let's say f of x looks like something like this. And I would say over the interval that I've drawn it, so it looks like from x is equal to 0, to maybe that point right over there, is this function continuous? Well you'd say no, it isn't. Look, we're 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. And you might even say we have a discontinuity at this value of x, right over here. We would call this a discontinuity. And actually this type of discontinuity is called a jump discontinuity. So you would say this is not continuous. It's obvious that 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, 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-- this is the discontinuity-- is called a removable discontinuity. Removable. One could make a reasonable argument that this also looks like a jump, but this is typically categorized as a removable discontinuity. Because if you just re-define 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 then 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 that I've depicted? And you would say, well, look. Yeah, it looks all connected all the way. There aren't any jumps over here. No removable discontinuities over here. This one looks continuous. And you would 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 rigorous definition of one. And since we already have a rigorous definition of limits, the epsilon delta definition, gives us a rigorous definition for limits. It's a 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. So let me draw an interval right over here. So it's between that x value and that x value. This is the x-axis, this is the y-axis. And let me draw my function over that interval. 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's not at the edge of my boundary. So this is an interior point for my interval. This would be an end point, and this would also be an end point. We'd say it's continuous at an interior point. So continuous at interior point, interior to my interval, means that the limit as, let's say at interior point c, so this is the point x is equal to c. We can say that it's continuous at the interior point c if the limit of our function-- this is our function right over here-- if the limit of our function as x approaches c is equal to the value of our function. Now does this make sense? Well, what we're saying is, is at that point, well this is f of 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 of c is right over there. That is f of c. Now is it the case that the limit of f of x, as x approaches c, is equal to f of c? Well, if we take the limit of f of x as x approaches c from the positive direction, it does look like it is f of c. It does look like it's equal to f of c. But if we take the limit-- but this does not equal the limit of f of x as x approaches c from the negative direction. As we go from the negative direction, we're not approaching f of c. So therefore, this does not hold up. In order for the limit to be equal to f of c, the limit from both the 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 this one is not continuous. What about this one right over here? And let me re-set it up. So let me make sure that that looks like a hole right over there. So we see here, what is the limit? The limit-- and this is our c, right over here-- the limit of f of x as x approaches c, let's say that that is equal to L. And 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 is that L does not equal f of c. This right over here is f of c. So once again, this would not pass our test. The limit of f of x as x approaches c, which is this right over here, is not equal to f of c. So once again, 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 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 at boundary-- or let me call it endpoint, actually, that would be better-- at endpoint c. So let's first consider just the left endpoint. If left endpoint-- so what I'm talking about, a left endpoint? Let me draw my axes, x-axis, y-axis. And let me draw my interval. So let's say this is the left endpoint of my interval, this is the right endpoint of my interval. And let me draw the function over that interval. Looks something like this. So when we talk about a left endpoint, we're talking 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 endpoint c-- that means that the limit f of x as x approaches c, well, we can't even approach see from the left hand side. We have to approach from the right. Is equal to f of c. And so this is really kind of a, we can only approach something from one direction. So we can't just say the limit in general, but we can say the limit from one side. So it's really 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 right over here. Which is the exact same thing as f of c. So we are continuous at that point. What's an example where an endpoint-- where we would not be continuous and an end point? 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 there. And then it would look something like that. Or there's no hole, the function just has a removable discontinuity right over there. At least visually it looks like that. And you see that 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 of c is up here. So f of 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 do we do if we're dealing with the right endpoint? So we say we're continuous at right endpoint c if-- so let me draw that, do my best attempt to draw it-- so this is my x-axis, this is my y-axis. Let me draw my interval that I care about, say it looks something like this. A right endpoint means c is right over there. And we can say that we are continuous at x, the function is continuous at x equals c means that the limit of f of 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, is equal to f of c. If we can say that, if this is true, then this implies that we are continuous at that right endpoint, c, and vice versa. And a situation where we're not? Well you could imagine instead of this being defined right at that point, you could create, you could say the function jumps up. Just like 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's kind of 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.