Why does the uniformly continuity and existence of right derivative implies the differentiabilty?

Question

Assume $f:\mathbb R\to \mathbb R$ is uniformly continuous function and has a right derivative $f'_+:\mathbb R\to \mathbb R$ which is also uniformly continuous. Where $f'_+$ is define by $$f'_+(x) = \lim_{h\to 0^+}\frac{f(x+h)-f(x)}{h}$$

Prove or disprove that $f$ is differentiable.