Let $f:\mathbb{R}^n\to\overline{\mathbb{R}}$ be convex and $x^*\in int(dom (f))$. If there exists a mapping $g:int(dom(f))\to \mathbb{R}^n$ satisfying
- $g(x)\in\partial f(x)$ for any $x\in int(dom(f))$
- $g$ is continuous at $x^*$
Then $\partial f(x^*) = \{\nabla f(x^*)\}$.
I know that there is a theorem stating that the conclusion is true when $f$ is convex differential. I tried to prove "the existence of $g$ implies $f$ is differentiable", but I am confused and cannot make reasonable inductions.
Please help me with the idea of this problem. Any help would be appreciated.
