If $f'$ is continuous, is then $f|_{[a,b]}$ with $a < b$ Lipschitz continuous?

Question

Let $f'$ be continuous. I am supposed to prove or disprove that $f|_{[a,b]}$ is Lipschitz continuous for all $a,b \in \mathbb{R}$ with $a < b$.

My intuition is that this is true and that I can use the central limit theorem for this, however, I am stuck on how to exactly do this.

Could anyone show me how to prove or disprove this?

Answer 1

Hints:

1. there is $c \ge 0$ such that $|f'(x)| \le c$ for all $x \in [a,b].$

2. let $u,v \in [a,b]$ and consider $|f(u)-f(v)|$. Invoke the mean value theorem.

Answer 2

Hint: If $f'$ is continuous, then its restriction to $[a,b]$ is bounded.