lean_workbook_plus_16403

Suppose that $f: \mathbb{R}\to \mathbb{R}$ is such that $|f(x)-f(y)|\le c|x-y|$ for all $x,y\in \mathbb{R}$ for some $c>0$ . Prove that $f$ is uniformly continuous. Can $f$ be differentiable everywhere? Provide an example if not. Characterize the differentiability of $f$ in terms of integrability.