Given a Lipschitz continuous function, prove there is some $M > 0$ such that $|f'(x)| < M$

Question

Given $f: I \rightarrow \mathbb{R}$ is Lipschitz continuous, where $I \in \mathbb{R}$ is an interval, prove there exists some $M > 0$ such that if $f$ is differentiable at a point $x \in (a,b)$, then $|f'(x)| < M$

Answer 1

Hint:

$$\left|\left|\frac{f(y)-f(x)}{y-x}\right| - |f'(x)|\right| \leqslant \left|\frac{f(y)-f(x)}{y-x} - f'(x)\right|,$$

and

$$|f'(x)| = \left|\lim_{y \to x} \frac{f(y)-f(x)}{y-x}\right| = \lim_{y \to x}\left| \frac{f(y)-f(x)}{y-x}\right|$$