Suppose that $h$ is continuous on $[a,b]$ and differentiable on $(a,b)$, and that $c \in (a, b)$. Suppose also that $\lim\limits_{x \to c} h'(x)$ exists. Prove that $h'$ is continuous at $c$.
I really have no idea how to think about this problem. I know if the limit exists then it's differentiable at the point, i was assuming differentiability implied continuity. Someone please give me advice.