Let $f$ be a real-valued function. Suppose I want to find a local maximum of $f$, but I decide to work with an ''approximation'' to $f$ --let us call it $g$. What is a suitable notion of ''approximation'' that gives conditions so that the critical points of $g$ approximate those of $f$, and such that the critical points of $g$ behave as those of $f$?



Question about optimizing a given function by optimizing an approximation
In general, small perturbations of the 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](https://en.wikipedia.org/wiki/%CE%93-convergence?wprov=sfla1)  which works for functions defined on topological spaces and its specialization to Banach spaces, which is [Mosco convergence](https://en.wikipedia.org/wiki/Mosco_convergence). 


