TITLE: Is a Lipschitz function differentiable?
QUESTION [22 upvotes]: Is a Lipschitz function differentiable?
I have been wondering whether or not this property applies to all functions. 

I do not need a formal proof, just the concept behind it. 
  Let $f: [a,b] \to [c,d]$ be a continuous function (What is more - it is uniformly continuous!) And let's assusme that it's also Lipschitz continuous on this interval. 

Does this set of assumptions imply that $f$ is differentiable  on $(a,b)$?

REPLY [37 votes]: It is not always true  indeed, good counterexample could be $x\mapsto |x-a|$. But rather, we have 

Theorem: Radamacher theorem says every Lipschitz function is almost everywhere differentiable

Fine a nice proof of this theorem here: An Elementary Proof of Rademacher's Theorem - James Murphy or Here using distribution theory