Let $f:\mathbb{R}^n\to \mathbb{R}\cup\{\infty\}$ be proper, lower semicontinuous and convex, and $\operatorname{dom}f=\{x\in \mathbb{R}^n\mid f(x)<\infty\}$ be the domain of $f$. Assume further that $f$ is strictly convex on $\operatorname{dom}f$. Then is the subgradient $\partial f(x)$ nonempty for all $x\in \operatorname{dom}f$?
The accepted answer in the post seems to suggest the assertion is correct. But the reference there assumes $\operatorname{dom}f=\mathbb{R}^n$. Could you refer me to a proof of the statement?