If a function $f: D \subset R^n \rightarrow R$ is convex, then for every $x,y \in D$ and $\alpha \in [0,1]$, $$ f(\alpha x + (1 - \alpha)y) \leq \alpha f(x) + (1 - \alpha)f(y). $$
I konw a convex function is a continuous function.
I just wonder if every convex function is differentiable amost every where.
This is important, because one of the most important way to solve a convex problem is gradient decent.