If $a,b$ are extended real numbers, $f$ is differentiable/$f'$ is continuous on $(a,b)$, prove uniform continuity.

Question

Suppose $a<b$ are extended real numbers and that $f$ is differentiable on $(a,b)$. Prove that if $f'$ is bounded on $(a,b)$ then $f$ is uniformly continuous on $(a,b).$

Answer 1

HINTS (sort of)

1. What does it mean that $f'$ is bounded?
2. What does it mean to be uniformly continuous?

Answer 2

Hint in addition to @mixedmath, if you know what those definitions mean: use the mean value theorem.

Answer 3

Apply Mean Value Theorem,and bound of $f$, $|f(x)-f(y)|=f'(c)|x-y|$