$f\colon (a,b) \to \mathbb{R}$ is continuous, with finite derivatives everywhere in $(a,b)$, except maybe at $c$.
If
$$\lim_{x\to c}f'(x) = B,$$
show that $f'(c)$ exists and equals $B$.
I'm not sure where to start on this. I've tried using the definitions of continuity and f' but it isn't working out
Well I started out with
$$\lim_{x\to c}\frac{f(x)-f(c)}{x-c} = f'(c)$$
is equivalent to saying
given $\epsilon\gt 0$ there exists $\delta\gt 0$ such that
$$|x-c| \lt \delta \Longrightarrow \left|\frac{f(x)-f(c)}{x-c} - f'(c)\right| \lt\epsilon.$$
I was trying to prove $f'(c)$ must exist since $f$ is continuous but I feel like I'm assuming what I'm trying to prove.