Is a function differentiable at a point if a certain limit exists

Question

So, I'm given the following situation:

Assume that $f$ is a continuous function on $(-1,1)$ and that $f$ is differentiable on $(-1,0)\cup(0,1)$. If $\lim_{x \to 0} f'(x)$ exists, then is $f$ differentiable at $x=0$?

I've tried to come up with counter-examples to no success! The more I think about this, the more I think this has to be right, but I haven't a clue on how to prove this!

Answer 1

Try applying the mean value theorem. If $x \not= 0$ is small enough, then for some $0 < |\xi| < |x|$ you have $$\frac{f(x) - f(0)}{x-0} = f'(\xi) \approx L$$ where $L = \lim_{x \to 0} f'(x)$.