If $f$ is Lipschitz of order $1$ over $[a,b]$, is it true that $f$ is differentiable over $[a,b]$?

Question

If $f$ is Lipschitz of order $1$ over $[a,b]$, is it true that $f$ is differentiable over $[a,b]$?

We are given that there is a constant $C$ such that $|f(x)-f(y)| \leq |x-y|$ for all $x$ and $y$ in $[a,b]$. I think there should be a counterexample to this since we can easily show a counterexample if our domain is $(-\infty,\infty)$. I think the same should be true here.

Answer 1

$f(x) = |x|$ is a counterexample.