Approximation of Continuous Functions by Lipschitz Functions

Abstract

Georganopoulos (see [1] ) has shown that a continuous function $% f:X\rightarrow B$, where $X$ is a compact metric space and $B$ a convex subset of a real normed space $Y$, is the uniform limit of a sequence of Lipschitz maps from $X$ to $B$. In this note we obtain a similar result, namely we show that a continuous function $f:X\rightarrow R$, where $X$ is a metric space, is a uniform limit of a sequence of locally Lipschitz maps from $X$ to $R$. When $X$ is compact and $B=Y=R$, we get the Georganopoulos's result.