Consider a closed, convex, proper function $g:\mathcal{X}\to\bar{\mathbb{R}}$ with $\mathcal{X}$ some reflexive Banach space (if it's easier, a Hilbert space is okay too) and $\bar{\mathbb{R}}$ is the real numbers with $+\infty$ and $-\infty$. Assume that there are two weakly convergent sequences $a_n\to a$ and $b_n\to b$ with
$$a_n\in\partial g(b_n)$$
for all $n$. When does it follow that $a\in \partial g(b)$? What if we had a nonreflexive Banach space or a Hilbert space instead?
I think $a\in\partial g(b)$ holds when $\partial g$ is an upper hemicontinuous correspondence. So maybe the better question is, when is $\partial g$ upper hemicontinuous?