Sal shows that if a function is continuous, the difference in the function's values approaches 0 as the difference in the x-values approaches 0. This is simply another way to define continuity.
- [Voiceover] The result that I hope to show you, or you give you an intuition for, in this video is something that we will use in the proof of the chain rule, or in a proof of the chain rule actually. We may do more than proof of the chain rule. But the result we're gonna look at is if we have some function u which is a function of x, and we know that it is continuous at x equals c, so if we know this, then that's going to imply that the change in u goes to zero as our change in x in this region around c goes to zero. This is what I want to get an intuition for. That if u is continuous at c, then as our change in x around c gets smaller, and smaller, and smaller as it approaches zero, then our change in u approaches zero as well. To think about this, or to even kind of prove it to ourselves a little bit more rigorously, let's think about what it means to be continuous at x equals c. Well the definition of continuity is, so this literally is the same thing as saying that the limit as x approaches c of u of x is equal to u of c. The limit that our function approaches as x approaches c is equal to the value of the function at c. We don't have a point discontinuity or a jump discontinuity. If we had a jump discontinuity, then the limit wouldn't exist, and we've seen that in previous videos. Now I'm just gonna manipulate this algebraically so it essentially gives us this conclusion right over here. So this we can rewrite. It's important to realize that u of c, this is just going to be some value. It looks like maybe this is a function of x or something, but no, this is just going to be some value. I've inputted c here and I've evaluated the function of that, and so this is going to be some number. It could be five, or seven, or pi, or negative one, but it's just going to be some value, some constant. So I can treat it like a constant. So this is going to be the same thing as saying the limit as x approaches c of u of x minus u of c is equal to zero. And actually, in the video where we prove that differentiability implies continuity, we started with this and we proved that right over there. We show that these two are equivalent things. But hopefully you could even think about the intuition. This is just if the limit of u of x is x approaches c is equal to this, then when you evaluate this limit, the limit is x approaches c, well this thing is going to approach u of c, because we saw it right up here, u of c minus u of c is indeed going to be equal to zero. So hopefully you don't feel like it's too much of a stretch, and you can just subtract u of c from both sides and apply properties of limits, and you can get this result as well. But this is interesting because this essentially can take us to this, the idea that as our change in x gets smaller, and smaller, and smaller as it approaches zero, then our change in our function is also going to approach zero. Now let's just graph this or visualize this to get a sense of that. So this is our x axis. Woops. That's our x axis. Let's call that our u axis maybe. And I did u intentionally because that's the variable we'll use in our proof of the chain rule video. Let's say this right over here is our function. Let's say this right over here, that is c. This right over here is u of c. U of c. Then let's just take some arbitrary x over here. So some arbitrary x, and then this right over here is u of x. So if we define our change, let me do this. If we define our change in u is equal to u of x minus u of c, which makes sense because this is our change in u. So let's say this is going to be u of x minus u of c. If we define our change in x is equal to x minus c, which it is in this case, it is x minus c, then we can rewrite this limit right over here. Instead of saying the limit as x approaches c, we could write the limit as delta x approaches zero, because if x approaches c, then delta x is going to approach zero. So we could write this the limit as delta x approaches zero of delta u is going to be equal to zero. We define this as our change in u, and it is our change in u. So this is equal to zero. So another way of thinking about this is as delta x approaches zero, our change in u, our change in the function, is going to approach zero. So as delta x approaches zero, delta u approaches zero. That's what we wrote over here. Delta u approaches zero as delta x approaches zero. In a lot of ways, this is hopefully common sense. We're dealing with a continuous function. As you get smaller and smaller, and you can just think of it this way, as you get smaller and smaller changes in x's, as our change in x gets smaller, and smaller, and smaller, and smaller, well it's because it's continuous. You wouldn't be able to say this for a discontinuous function, but because it's continuous, or you wouldn't be able to say this for some discontinuous functions, as our change in x gets smaller, and smaller, and smaller, then our change in u is going to get smaller, and smaller, and smaller. So it makes intuitive sense, but hopefully this makes you feel even better about it because we're going to use this idea to prove the chain rule in the next video.