The result that I hope to show you or 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 one 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, 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, or it gets smaller and smaller and smaller as it approaches zero, then our change in u approaches zero as well. So let's just, to think about this, or 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. That 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 going to manipulate this algebraically, so it essentially gives us this conclusion right over here.