Uniformly continuous derivative

Question

Let $f:[a,b)\to\mathbb{R}$ be a differentiable function such that $f$ and $f'$ are uniformly continuous in $(a,b)$. Is it true then that $f'$ is continuous at a?

Note: By $f'(a)$ I mean $\lim_{h\to0^{+}}\frac{f(a+h)-f(a)}{h}$.

Answer 1

Hints:

(1) Show that $f'(x_n)$ is a Cauchy sequence when $x_n \to a+$. Show $\lim_{x \to a+} f'(x) = L $ exists uniquely.

(2) Use the mean value theorem

$$ \frac{f(x+h) - f(x)}{h} = f'(\xi)$$