Let $f$ be a convex function from $\Bbb R^n$ to $\Bbb R$ and $x\in \mathbb R^n$
Suppose $f$ has derivatives in every direction in $x$.
Prove that $f$ is differentiable at $x$.
I already proved that a convex function is continuous and locally-Lipschitz, but I don't see how this can help here.
I know the statement is not true without the assumption that $f$ is convex, see for example this question: $f$ not differentiable at $(0,0)$ but all directional derivatives exist
Since the function is convex, I guess something can be done using subdifferentials.