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Let $(X,d)$ be a locally compact metric space. Consider the space of complex continuous functions $C_0(X)$ which vanish at infinity.

I am interested in the following fact:

The space of compactly supported Lipschitz functions $Lip_{c,d}(X)$ is uniformly dense in $C_0(X)$.

I know that $Lip_d(X)$ is uniformly dense in $C_0(X)$, and that $C_c(X)$ is uniformly dense in $C_0(X)$.

  1. But how can one prove that $Lip_{c,d}(X)$ is uniformly dense in $C_c(X)$?
  2. Is there any reference for the above fact?

Many thanks!

Curious
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  • This follows directly from the triangular inequality. First approximate $f \in C_0(X)$ by a function $g \in C_c(X)$ and then approximate $g$ by a Lipschitz continuous function on the support of $g$. – Klaus Jul 29 '20 at 13:06
  • @Klaus, Hi! And what would be a Lipschitz continuous function on the support of $g$? We may have that the Lipschitz function which unif. approx. $g$ has no compact support and just vanishes at infinity. – Curious Jul 29 '20 at 14:23
  • In the second step you use the result on $Y = \mathrm{supp}(g)$ (as opposed to $X$), so that the approximation to g will also have support in $Y$. – Klaus Jul 30 '20 at 12:54

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