Continuity at a point | Limits and continuity | AP Calculus AB | Khan Academy

What we're going to do in this video is come up with a more rigorous definition for continuity and the general idea of continuity. We've got an intuitive idea of the past; that a function is continuous at a point is if you can draw the graph of that function at that point without picking up your pencil. So what do we mean by that?

This is a very, this what I just said is not that rigorous, or not rigorous at all. Well, let's think about the point right over here. Let's say that's c. If I can draw the graph at that point, the value of the function at the point, without picking up my pencil or my pen, then it's continuous there. So I could just start here, and I don't have to pick up my pencil, and there you go, I can draw, I can go through that point. So we could say that our function is continuous there.

But if I had a function that looked somewhat different than that, if I had a function that looked like this—let's say that it is defined up until then, and then there's a bit of a jump, and then it goes like this—well, this would be very hard to draw with. This would be, this function would be very hard to draw going through x equals c without picking up my pen. Let's see, my pen is touching the screen, touching the screen, touching the screen. How do I keep drawing this function without picking up my pen? I would have to pick it up and then move back down here.

That is an intuitive sense that we are not continuous in this case right over here. But let's actually come up with a formal definition for continuity and then see if it feels intuitive for us. So the formal definition of continuity—let's start here. We'll start with continuity at a point. So we could say the function f is continuous at x equals c if and only if—I'll draw this two-way arrow to show if and only if—the two-sided limit of f of x as x approaches c is equal to f of c.

So this seems very technical, but let's just think about what it's saying. It's saying, look, if the limit as we approach c from the left and the right of f of x, if that's actually the value of our function there, then we are continuous at that point. So let's look at three examples. Let's look at one example where we're by our picking up the pencil idea; it feels like we're continuous at a point. Then let's think about a couple of examples where it doesn't seem like we're continuous at a point and see how this more rigorous definition applies.

So let's say that my function—so let's say this right over here is y is equal to f of x, and we care about the behavior right over here when x is equal to c. This is my x-axis, that's my y-axis, so we care about the behavior when x is equal to c. Notice from our first intuitive sense, I can definitely draw this function as we go through x equals c without picking up my pencil, so it feels continuous there.

There's no jumps or discontinuities that we can tell; it just kind of keeps on going. It seems all connected is one way to think about it, but let's think about this definition. Well, the limit as x approaches c from the left looks like it is approaching f of c, so this is the value f of c right over here. As we approach from the right, it also looks like it's approaching f of c, and we are defined right at x equals c, and it is the value that we are approaching from both the left or the right.

So this seems good in this scenario. Now let's look at some scenarios that we would have to pick up the pencil as we draw the function through that point when x is equal to c. Let's look at a scenario, let's look at a scenario where we have what's often called a point discontinuity. Although you don't have to know at this point—not, no pun intended—the formal terminology for it.

So let's say we have a function that, let's see, this is c, and let's say our function looks something like this. So we go like this, and let's say it's equal to that. So f of c is right over here; f of c would be that value. But what's the limit as x approaches c? So the limit as x approaches c, and this would be a two-sided limit of f of x, well, this is as we approach from the left, it looks like we are approaching this value right over here, and from the right, it looks like we are approaching that same value.

We could call that l, and l is different than f of c. So in this case, by our formal definition, we will not be continuous at f. We will not be continuous for x is or at the point x when x is equal to c. You can see that there; if we tried to draw this—okay, my pencil is touching the paper, touching the paper, touching the paper—uh-oh! If I needed to keep drawing this function, I'd have to pick up my pencil, move it over here, then pick it up again, and then jump right back down.

But this rigorous definition is giving us the same conclusion: the limit as we approach x equals c from the left and the right is a different value than f of c, and so this is not continuous. Let's think about another scenario. Let's think about a scenario—and actually, maybe let's think about a scenario where the limit, the two-sided limit, doesn't even exist.

So there are my axes: x and y. Let's say it's doing something like this. Let's say it's doing something like this, and then it does something like this and goes like that. Let's say that this right over here is our c, and so let's see—this is f of c right over here, that is a little bit neater, that is f of c. It does look like the limit as x approaches c from the left—so from values less than c, it does look like that is approaching f of c.

But if we look at the limit as x approaches c from the right, that looks like it's approaching some other value. That looks like it's approaching this value right over here; let's call it l—it's approaching l. And l does not equal f of c. So in this situation, the two-sided limit doesn't even exist. We're approaching two different values when we approach from the left and from the right.

Since the limit doesn't even exist at c, this is definitely not going to be continuous. This matches up to our expectations with our little "can we have to pick up the pencil" test. If I have to draw this, I can leave my pencil; it's on the paper, it's on the paper, it's on the paper, it's on the paper. How am I going to continue to draw this function, this graph of the function, without picking up my pencil? Pick it up, put it back down, and then keep drawing it.

So once again, this right over here is not continuous, both intuitively by our pick-up-the-pencil definition, and also by this more rigorous definition where, in this case, the limit, the two-sided limit at x equals c doesn't even exist. So we're definitely not continuous. But even when the two-sided limit does exist, if the limit is a different value than the value of the function, that will also not be continuous.

The only situation that is going to be continuous is if the two-sided limit approaches the same value as the value of the function, and if that's true, then we're continuous. If we're continuous, that is going to be true.