# lipschitz function in a compact, is it differentiable?

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)|)

Can we prove that $$f$$ is differentiable ?

• Nope. A counterexample is $f(x)= |x|$. See en.wikipedia.org/wiki/Rademacher%27s_theorem for a similar result. – Crostul Jan 18 at 16:28
• @Crostul yeh you re right, thank you – Anas BOUALII Jan 18 at 16:29
• @Crostul minor correction: $|x|$ is differentiable on $[0,1].$ – zhw. Jan 18 at 19:22

No. For example, try $$f(x) = |x - 1/2|$$.
In general, no. The function $$x \mapsto |x|$$ gives a counterexample.