Differentiability

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).