Functions with a dense set of proper local maximum points

Abstract:

Let $X$ be any metric space. The existence of continuous real functions on $X$, with a dense set of proper local maximum points, is shown. Indeed, given any $\sigma$-discrete set $S \subset X$, the set of all $f \in C(X)$, which assume a proper local maximum at each point of $S$, is a dense subset of $C(X)$. This implies, for a perfect metric space $X$, the density in $C(X,Y)$ of "nowhere constant" continuous functions from $X$ to a normed space $Y$. In this way, two questions raised in [2] are solved.