A Note on Convergence of Level Sets

Given a sequence of functions f(n) converging in some topology to a function f, in general the 0-level set of f(n) does not give a good approximation of the one of f. In this paper we show that, if we consider an appropriate perturbation of the 0-level set of f(n), we get a sequence of sets converging to the 0-level set of f, where the type of set convergence depends on the type of convergence of f(n) to f.