A quite flexible framework for this typeIn general, small perturbations of problemsthe objective may change the set of local maxima drastically. Just think of a flat local maximum and adding a small wiggling (so also uniform approximation does not really help).
However, there is a notion of convergence of functions, that is build in a way to ensure convergence of extreme points and this is the so-called Gamma convergence which works for functions defined on topological spaces and its specialization to Banach spaces, which is Mosco convergence.