Assume that $g\colon\mathbb{R}^n\to\left(-\infty,\infty\right]$ is continuous over its domain, convex, closed and proper. For some $x,y\in\mathbb{R}^n$, assume that $x\in\partial g\left(y\right)$. Also, suppose that there exists a sequence $\left\{x^k\right\}$ such that $x^k\to x$.
Can we say that there exists a sequence $z^k\in\partial g\left(x^k\right)$ such that $z^k\to y$? Are there any assumptions need to made on the function $g$ for this statement to be true? Should $g$ be LSC? Something else?
I encountered this problem while trying to prove convergence of an algorithm to a critical point. Any help would be much appreciated.