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



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> 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.
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Approximation by locally Lipschitz functions
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 on a convex set of a normed space 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).


