Theorem of the Maximum for discrete sequences of constraint sets?

Question

Suppose that $\{X_{n}\}_{n=1}^{\infty}$ is a sequence of sets that converges to $X$ in some sense. Let $f$ be a real-valued function.

I am interested in conditions under which

$$ \lim_{n \rightarrow \infty} \max_{x \in X_{n}} f(x) = \max_{x \in X} f(x)$$

This seems reminiscent to Berge's Theorem of the Maximum, but different in that it involves a discrete sequence of sets.

Where should I look for conditions that ensure the above convergence? Is there a name for problems like these? Recommended references?

Answer 1

I believe that the right notion is epi-convergence.