Differentiability and continuity | Derivatives introduction | AP Calculus AB | Khan Academy

What we're going to do in this video is explore the notion of differentiability at a point. That is just a fancy way of saying, does the function have a defined derivative at a point?

So let's just remind ourselves of a definition of a derivative. There are multiple ways of writing this. For the sake of this video, I'll write it as the derivative of our function at point c. This is Lagrange notation. With this f prime, the derivative of our function f at c is going to be equal to the limit as x approaches c of f of x minus f of c over x minus c.

At first, when you see this formula, and we've seen it before, it looks a little bit strange. But all it is, is it's calculating the slope. This is our change in the value of our function, or you could think of it as our change in y if y is equal to f of x. This is our change in x, and we're just trying to see, well, what is that slope as x gets closer and closer to c? As our change in x gets closer and closer to zero, we talk about that in other videos.

So I'm now going to make a few claims in this video. I'm not going to prove them rigorously; there's another video that will go a little bit more into the proof direction. But this is more to get an intuition.

The first claim that I'm going to make is: if f is differentiable at x equals c, then f is continuous at x equals c. So I'm saying if we know it's differentiable, if we can find this limit, if we can find this derivative at x equals c, then our function is also continuous at x equals c.

It doesn't necessarily mean the other way around. Actually, we'll look at a case where it's not necessarily the case that if you're continuous, then you're definitely differentiable. But another way to interpret what I just wrote down is: if you are not continuous, then you definitely will not be differentiable. If f is not continuous at x equals c, then f is not differentiable at x equals c.

So let me give a few examples of a non-continuous function and then think about whether we would be able to find this limit. The first is where you have a discontinuity; our function is defined at c, it's equal to this value, but you can see as x becomes larger than c, it just jumps down and shifts right over here.

So what would happen if you were trying to find this limit? Well, remember, all this is, is a slope of a line between when x is some arbitrary value. Let's say it's out here; so that would be x. This would be the point (x, f of x), and then this is the point (c, f of c) right over here.

So this is (c, f of c). If you find the left-sided limit right over here, you're essentially saying, okay, let's find this slope. Then let me get a little bit closer, and let me find this slope. Then let's get x even closer than that and find this slope.

In all of those cases, it would be zero; the slope is zero. So one way to think about it is the derivative, or this limit as we approach from the left, seems to be approaching zero. But what about if we were to take x's to the right?

So instead of our x's being there, what if we were to take x's right over here? Well, for this point (x, f of x), our slope, if we take (f of x - f of c) over (x - c), that would be the slope of this line. If we get x to be even closer, let's say right over here, then this would be the slope of this line. If we get even closer, then this expression would be the slope of this line.

As we get closer and closer to x being equal to c, we see that our slope is actually approaching negative infinity. Most importantly, it's approaching a very different value from the right. This expression is approaching a very different value from the right than it is from the left.

So in this case, this limit up here won't exist. We can clearly say this is not differentiable. So once again, not a proof here; I'm just getting an intuition for if something isn't continuous, it's pretty clear, at least in this case, that it's not going to be differentiable.

Let's look at another case; let's look at a case where we have what's sometimes called a removable discontinuity or a point discontinuity.

So once again, let's say we're approaching from the left. This is x; this is the point (x, f of x). Now what's interesting is, where as this expression is the slope of the line connecting (x, f of x) and (c, f of c), which is this point, not that point. Remember we have this removable discontinuity right over here.

So this would be this expression is calculating the slope of that line. If x gets even closer to c, well then we're going to be calculating the slope of that line. If x gets even closer to c, we're going to be calculating the slope of that line.

As we approach from the left, as x approaches c from the left, we actually have a situation where this expression right over here is going to approach negative infinity. And if we approach from the right, if we approach with x larger than c, well this is our (x, f of x). So we have a positive slope, and then as we get closer, it gets more positive, more positive, approaches positive infinity.

But either way, it's not approaching a finite value, and one side is approaching positive infinity while the other side is approaching negative infinity. This limit of this expression is not going to exist.

So once again, I'm not doing a rigorous proof here, but try to construct a discontinuous function where you will be able to find this. It is very, very hard. You might say, well, what about the situations where f is not even defined at c?

Which for sure you're not going to be continuous if f is not defined at c. Well, if f is not defined at c, then this part of the expression wouldn't even make sense, so you definitely wouldn't be differentiable.

But now let's ask another thing. I've just given you good arguments for when you're not continuous; you're not going to be differentiable. But can we make another claim that if you are continuous, then you definitely will be differentiable?

Well, it turns out that there are for sure many functions, an infinite number of functions that can be continuous at c but not differentiable. So for example, this could be an absolute value function. It doesn't have to be an absolute value function, but this could be y is equal to the absolute value of (x - c).

And why is this one not differentiable at c? Well, think about what's happening. Think about this expression. Remember this expression? All it's doing is calculating the slope between the point (x, f of x) and the point (c, f of c).

So if x is say out here, this is (x, f of x). It's going to calculate, and as we're taking the limit for as x approaches c from the left, we'll be looking at this slope. Then as we get closer, we'll be looking at this slope, which is actually going to be the same. In this case, it would be -1.

So as x approaches c from the left, this expression would be -1. But as we uniformly approach c from the right, this expression is going to be 1. The slope of the line that connects these points is 1.

The slope of the line that connects these points is also 1. So the limit of this expression, or I would say the value of this expression is approaching two different values as x approaches c from the left or the right. From the left, it's approaching -1, or it's constantly -1.

From the right, it's 1, and it's approaching 1 the entire time. So we know if you're approaching two different values from the left-sided or the right-sided limit, then this limit will not exist.

So here, this is not differentiable. Even intuitively, we think of the derivative as the slope of the tangent line. You can actually draw an infinite number of tangent lines here. It's one way to think about it. You could say, well, maybe this is the tangent line right over there, but then why can't I make something like this the tangent line that only intersects at the point (c, 0)?

Then you could keep doing things like that. Why can't that be the tangent line? You could go on and on and on. The big takeaways here, at least intuitively, in a future video, I'm going to prove to you that if f is differentiable at c, then it is continuous at c.

This can also be interpreted as: if you're not continuous at c, then you're not going to be differentiable. These two examples will hopefully give you some intuition for that.

But it's not the case that if something is continuous, then it has to be differentiable. It oftentimes will be differentiable, but it doesn't have to be differentiable. This absolute value function is an example of a continuous function at c, but it is not differentiable at c.