Proof: Differentiability implies continuity | Derivative rules | AP Calculus AB | Khan Academy

Sal shows that if a function is differentiable at a point, it is also continuous at that point.

AP Calculus AB on Khan Academy: Bill Scott uses Khan Academy to teach AP Calculus at Phillips Academy in Andover, Massachusetts, and heÕs part of the teaching team that helped develop Khan AcademyÕs AP lessons. Phillips Academy was one of the first schools to teach AP nearly 60 years ago.

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[arunkhanna]

Differentiability implies Khantinuity!!!

[emresarac]

I wish my teacher taught that like this way. He just goes too fast that we cannot follow what he's doing xd. Thanks a lot Khan Academy!

[cansucorbac]

"The AP Calculus course doesn't require knowing the proof of this fact, but we believe that as long as a proof is accessible, there's always something to learn from it. In general, it's always good to require some kind of proof or justification for the theorems you learn." :)))) Canım.

[booguy]

Why do you start the proof with lim x→c (f(x) - f(c))?

[giack]

Hello and thank you for the video, but I think there's an error in 9:44: If you have lim (x -> c) (f(x) - f(c)) = 0, you can conclude that this is equivalent to: lim(x -> c) f(x) = lim(x -> c) f(c) only if you know that the single limits exist, but you don't know yet if lim(x -> c) f(x) actually exist!

[serve]

I need to learn this 4 times for it to get inside my head

[mehulkathiriya]

awesome...I like it even more than iit lecture.short and simple....

[arunkumaru]

Why you took lim( f(x)-f(c)) ??
x>c

[rainerwahnsinn]

You are only allowed to split the limit, if both parts are *convergent*!

[stickmanbattle]

This video only proves limx->c(f(x)) = f(c), but the most important info if the lim is not continuous at that point it means is not differentiable.

[wagsman]

Excellent. I suppose you can assume f(c) exists, because if it didn't the derivative would fail.

[jaswanthramesh]

can f'(C) be infinity?(happens for a vertical straight line), if it is infinity, then 0*infinity becomes an indeterminate form . Then we wont get this.

[lizmurith]

Why do you multiply and divide by (x-c)? Why are we changing the original formula?

[Dyslexic-Artist-Theory-on-Time]

Can we think of this as a process over a period of time? 

[puspitapaul]

please explain the differentiability of |log|x||

[CMDF]

I think I missed something here. Why can we say that the limit of f(c) as x approaches c is equal to f(c)? I don't see where we proved that anywhere. I don't understand where it comes from. I must have missed some detail. I understand how we proved that the limit of f(x)-f(c) as x approaches c = 0, but if our proof is using the fact that f is continuous at c, isn't that circular logic? Or does the limit of f(x)-f(c) as x approaches c = 0 somehow require f to be continuous at c? I'm so lost :(

[majorgeneralrahul]

Fun Fact :- Even if you assume it's NOT differentiable you'll get the same answer if you use this method. :D

[renayachan]

Does it mean that if a function aint continuous at a number it is also not differentiable at that number?

[kamitube]

I lost at 7:10, can anybody please help me? why is he multiplying (x-c) at top and bottom?

[aragonification]

But if differentiability at a point means that the derivative of that point exists, why the first example from the right side is not continuous?