Approximation by locally Lipschitz functions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T09:02:55Z http://mathoverflow.net/feeds/question/43625 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/43625/approximation-by-locally-lipschitz-functions Approximation by locally Lipschitz functions ε-δ 2010-10-26T03:34:05Z 2010-10-26T10:35:02Z <p>Could you tell me what is the name and/or reference for the following theorem:</p> <blockquote> <p>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.</p> </blockquote> http://mathoverflow.net/questions/43625/approximation-by-locally-lipschitz-functions/43633#43633 Answer by Will Jagy for Approximation by locally Lipschitz functions Will Jagy 2010-10-26T04:48:00Z 2010-10-26T04:48:00Z <p>In that case, take a look at:</p> <p>Lipschitz-type functions on metric spaces</p> <p>M. Isabel Garrido, Jesús A. Jaramillo</p> <p>Journal of Mathematical Analysis and Applications</p> <p>Volume 340, Issue 1, 1 April 2008, Pages 282-290</p> <p>Abstract</p> <p>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.</p> <p>Keywords: Lipschitz functions; Banach–Stone theorem; Uniform approximation</p> http://mathoverflow.net/questions/43625/approximation-by-locally-lipschitz-functions/43653#43653 Answer by Pietro Majer for Approximation by locally Lipschitz functions Pietro Majer 2010-10-26T10:29:25Z 2010-10-26T10:35:02Z <p>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.</p> <p>A close result is that <em>uniformly continuous</em> ($\mathbb{R}$-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). </p>