existing limit and differentiability

Question

Let $f: (a, b) \to \mathbb R$ continuous function that is differentiable in every $x \in (a, b)$ except $x_0$. Show that if there exists a finite limit

$L = \lim_{x \to x_0} f´(x)$

then $f$ is differentiable in $x_0$ and $f´(x_0) = L$

I think I have a kind of a hunch here, maybe by using MVT? But I'm not sure how to put it.

Answer 1

HINT:

From the mean value theorem, for $a

Take the limit as $x\to x_0$ from the right. The limit is $L$. Now, show that the limit from the left exists.