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 ?
No. For example, try $f(x) = |x - 1/2|$.
In general, no. The function $x \mapsto |x|$ gives a counterexample.
However, Lipschitz functions are differentiable almost everywhere:
Also note that if the derivative is bounded, the converse is true.