TITLE: Approximation by locally Lipschitz functions
QUESTION [9 upvotes]: 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.

REPLY [7 votes]: oh what a good question it is ! i think this question is from the following paper :
see : https://projecteuclid.org/download/pdf_1/euclid.rae/1230939175
and your question is the same with theorem 2 in this paper !
is it helpful ? thank you !