Approximation by locally Lipschitz functions - MathOverflow

Approximation by locally Lipschitz functions

2010-10-26T03:34:05Z

Could you tell me what is the name and/or reference for the following theorem:

Let $M$ be a metric space. Then any continuous function $f:M\to\mathbb R$ can be a be uniformly approximated by a locally Lipschitz functions.

Answer by Will Jagy for Approximation by locally Lipschitz functions

2010-10-26T04:48:00Z

In that case, take a look at:

Lipschitz-type functions on metric spaces

M. Isabel Garrido, Jesús A. Jaramillo

Journal of Mathematical Analysis and Applications

Volume 340, Issue 1, 1 April 2008, Pages 282-290

Abstract

In order to find metric spaces X for which the algebra Lip*(X) of bounded Lipschitz functions on X determines the Lipschitz structure of X, we introduce the class of small-determined spaces. We show that this class includes precompact and quasi-convex metric spaces. We obtain several metric characterizations of this property, as well as some other characterizations given in terms of the uniform approximation and the extension of uniformly continuous functions. In particular we show that X is small-determined if and only if every uniformly continuous real function on X can be uniformly approximated by Lipschitz functions.

Keywords: Lipschitz functions; Banach–Stone theorem; Uniform approximation

Answer by Pietro Majer for Approximation by locally Lipschitz functions

2010-10-26T10:29:25Z

Actually the uniform density of locally Lipschitz functions is quite an immediate consequence of the paracompactness of metric spaces (Stone's theorem), and of the fact that, of course, metric spaces admit locally Lipschitz partitions of unity. Note that this way you also have the general result for Banach-valued functions, that is, with a given Banach space as a codomain.

A close result is that uniformly continuous ($\mathbb{R}$-valued) functions can be uniformly approximated by (uniformly) Lipschitz functions. In this case, an explicit approximation for a function $f$ is obtained just taking $f_k:=$ the infimum of all $k$-Lipschitz functions above $f.$ Then $f_k$ is k-Lipschitz and $f_k\to f$ uniformly as $k\to \infty$ (moreover, the uniform distance of $f$ and $f_k$ can be evaluated in terms of the modulus of continuity of $f$), without need of Stone's theorem. I think that variant of this construction should work for locally Lipschitz approximation of continuous functions (always in the scalar-valued case).