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 \subseteq 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.
Functions with a dense set of proper local maximum points
VILLANI, Alfonso
1985-01-01
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 \subseteq 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.File in questo prodotto:
Non ci sono file associati a questo prodotto.
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.