Continuity of derivative function

Question

Let $f:X \to Y$ be a function such that it is differentiable on the interval $[x ,y] \subseteq X (x < y).$ If $a \in (x , y)$ and if $\lim\limits_{z \to a^+}f'(z)$ and $\lim\limits_{z \to a^-}f'(z)$ both exists, then $f'$ is continuous at $a$.

I tried using Darboux's Theorem and the property of limit but was not able to bring a complete proof. Is there any hints to tackle this?

Answer 1

Argue by contradiction. WLOG, assume $\lim_{y\uparrow a}f'(y)=m<f'(a)$. By intermediate value property, for any $n>0$ we can find point $x_n\in(\max(a-\frac1n,x),a)$ with $f'(x_n)=\frac{m+f'(a)}{2}$. So $\lim_n f'(x_n)=\frac{m+f'(a)}{2}\neq m=\lim_{y\uparrow a}f'(y)$, but $x_n\uparrow a$ by construction.