lipschitz function in a compact, is it differentiable?

Question

Let $f$ be a lipschitz function in $[0,1]$,

(it exists a $C>0$ that we have for all $x,y \in [0,1]$ $|f(x)-f(y)|<C.|x-y|$)

Can we prove that $f$ is differentiable ?

Answer 1

No. For example, try $f(x) = |x - 1/2|$.

Answer 2

In general, no. The function $x \mapsto |x|$ gives a counterexample.

However, Lipschitz functions are differentiable almost everywhere:

Lipschitz continuity implies differentiability almost everywhere.

Also note that if the derivative is bounded, the converse is true.