What's the definition of a point being differentiable?

[Treadstone 71]

"Suppose f is continuous on [a,b] and c in (a,b). Suppose f is differentiable at all points of (a,b) except possibly at c. Assume further that lim(x->c)f'(x) exists and is equal to k. Prove that f is differentiable at c and f'(c)=k"

Since the lim f'(x) as x->c exists, f'(c) either equals k, exists but doesn't equal to k, or undefined. I showed that if it is defined, it must equal to k by using the intermediate value property of f'. But I can't show that f'(c) has to be defined. I tried contradiction, saying if f'(c) is undefined, but I can't run into a contradiction.

 

[Zurtex]

What's the definition of a point being differentiable?

 

[Treadstone 71]

lim(x->c) (f(x)-f(c))/(x-c) = f'(c), provided the limit exists.

 

[HallsofIvy]

Try using the mean value theorem on (x₀, c) and (c, x₁) to show that the left and right limits of the difference quotient exist and are the same.

 

[Treadstone 71]

I'm not sure what you mean by difference quotient. The mean value theorem assumes the existence of f'(c).