Let's assume that $f:\mathbb{R}\to\mathbb{R}$ is differentiable and its derivative $f'(x)$ is not continuous. Applying the mean value theorem we get: $$ \frac{f(x)-f(a)}{x-a}=f'(\xi), \text{ where } a<\xi<x. $$
Now the claim is that $f'(\xi(x))$ is continuous.
At each point $x\neq a$ this follows directly from the rules of continuous functions. If we set $f'(\xi(a)):=f'(a)$ and choose a sufficiently small $\delta>0$ then for all $x$ with $|x-a|<\delta$ we see that: $$ \left|\frac{f(x)-f(a)}{x-a}-f'(a)\right|<\epsilon, \text{ because }f \text{ is differentiable}. $$ Hence, $f'(\xi(x))$ is continuous.
Further, we see that $x\to a\implies \xi\to a$. So $\lim\limits_{x\to a} f'(\xi(x))=\lim\limits_{\xi\to a} f'(\xi) = f'(a)$. How is this possible if $f'(x)$ is not continuous?