I've been stating at this problem for a while and have no idea where to even start. Not looking for an answer, but really just a nudge in the right direction.

The homework is in a section covering Rolle's Theorem and the Mean Value Theorem.

Suppose $\displaystyle f$ is differentiable on $\displaystyle (a,b)$ except possibly at $\displaystyle x_o \in (a,b)$ and is continuous on $\displaystyle [a,b] $; assume $\displaystyle lim(x \rightarrow x_o) f'(x)$ exists. Prove that $\displaystyle f$ is differentiable at $\displaystyle x_o$ and $\displaystyle f'$ is continuous at $\displaystyle x_o$.