I was involved into this subject when I answered this question from MSE. Trying to generalize my answer, I am thinking about a following

*Question.* Let $X$ and $Y$ be metric spaces. When each continuous function
from $X$ to $Y$ can be approximated by Lipschitz functions
in the topology of uniform convergence on compact sets?

It seems that I can prove that the question has a positive answer provided a pair $(X,Y)$ satisfies the following

*Condition.* Each bounded (that is with finite diameter of the image) function from $A$ to $Y$ defined on a uniformly discrete subset $A$ of the space $X$
can be extended to a Lipschitz function from $X$ to $Y$. (I call a subset $A$ of a metric space
$(X,d)$ *uniformly discrete,* provided there exists a number $\varepsilon>0$ such that
$d(x,y)\ge\varepsilon$ for each distinct points $x,y\in A$.)
In particular, Valentine Theorem (see, [p. 11, LR] or [V]) implies that
a pair $(X,Y)$ satisfies Condition provided $X$ and $Y$ are Hilbert spaces.

We found no references on the above results and I think about a publication and a continuation of this investigation. But I am not a specialist in this domain.

So, I care about the following questions. *Are the above results new, good
and worthy to be published somewhere? What another related problems are worthy to be investigated?
In particular, is it known which pairs $(X,Y)$ of metric spaces satisfies Condition?*

Thanks.

*References*

[LR] Ronnie Levy, Michael David Rice.
*The approximation and extension of uniformly continuous Banach space valued
mappings*, Commentationes Mathematicae Universitatis Carolinae,
**24**:2 (1983), 251--265.

[V] F.A. Valentine, *A Lipschitz condition preserving extension for a vector function,* Amer. J.
Math., **67** (1945), 83--93.

*Related questions*

Approximation by locally Lipschitz functions

explicit extention of Lipschitz function (Kirszbraun theorem)