As proven here for example, a continuously differentiable function is locally lipschitz.
But does the converse hold? Is there a function (maybe even on $\mathbb R$) that is locally lipschitz but not continuously differentiable?
I thought of $f:\mathbb R\to\mathbb R, x\mapsto |x|$ which satisfies the lipschitz condition for $L=2$ (even globally) but it is not differentiable at $0$.