$K \subset \mathbb{R}$ is a compact set such that $ K=K'$ and $f:K \rightarrow \mathbb{R}$ is a function of class $C^1$, then $f$ is Lipschitzian.

Question

Prove that if is a $\emptyset \ne K \subset \mathbb{R}$ is a compact set such that $ K=K'$ and $f:K \rightarrow \mathbb{R}$ is a function of class $C^1$, then $f$ is Lipschitzian.

Can anyone help me with suggestions?

Answer 1

A standard result says that any function from an interval $f\colon I \to \mathbb{R}$ with bounded first derivative is Lipschitz. Now, $f$ being $C^1$ simply shows that $f'\colon K \to \mathbb R$ actually exists and it is continuous, and $K$ being compact lets us use the Extreme Value Theorem on $f'$ to conclude that it is bounded. Thus, the initial remark tells that $f$ is indeed Lipschitz.