# Continuity of the derivative at a point given certain hypotheses [duplicate]

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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.

## marked as duplicate by Hans Lundmark, Jonas Dahlbæk, user370967, Frits Veerman, kingW3Jun 29 '17 at 13:49

• Yes, differentiability of $h$ implies continuity of $h$. But you're asked to prove the continuity of $h'$ at $c$, not of $h$. – user49640 Apr 6 '17 at 2:57