Functions with a dense set of proper local maximum points

Abstract

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