In the smooth setting, Whitney's approximation theorem says the following: If $M,N$ are smooth manifolds and $f,g:M\to N$ are smooth functions that are continuously homotopic (ie there is a continuous homotopy $H:M\times [0,1]\to N$ between them) then they are also smoothly homotopic (ie there exists a smooth homotopy $\tilde{H}$ between them).


Does something similar hold for Lipschitz functions between Lipschitz manifolds? More precisely, **if $M,N$ are two Lipschitz manifolds and $f,g:M\to N$ are Lipschitz functions that are continuously homotopic, is it true that they are also Lipschitz homotopic (ie there exists a Lipschitz homotopy between them)?**



Whitney's approximation theorem for Lipschitz manifolds
For compact Lipschitz manifolds this follows from the main result of the paper
by Liu, Luofei, Yu, Hanfu,
Liu, Ye
"[Converting uniform homotopies into Lipschitz homotopies via moduli of
continuity.](https://www.sciencedirect.com/science/article/abs/pii/S0166864120303205)" Topology Appl. 285 (2020)


But the above paper is surely an overkill. It gives quantitative bounds on the homotopy and qualitative result should be known much earlier.


I expect it's known that for a compact Lipschitz manifold $N$, for some Lipschitz embedding of $N$ into some $\mathbb R^n$ a small neighbourhood of $N$ Lipschitz retracts onto $N$. This would easily imply the result.


It's likely even true that $N$ is a Lipschitz ANR.


It's also clear that there is an appropriate version of this for noncompact manifolds too.


Edit: I found a reference to the fact that any Lipschitz manifold is a Lipschitz ANR. This immediately follows from Theorem 5.1 in [this paper](http://www.acadsci.fi/mathematica/Vol03/vol03pp085-122.pdf).


Therefore what I said above works. It is enough to show that any two sufficiently $C^0$ close Lipschitz maps $f,g:M\to N$ are Lipschitz homotopic.
Embed $N$ by a Lipschitz map into a Euclidean space $\mathbb R^n$, connect $f$ to $g$ by straight line homotopy in $\mathbb R^n$ and then use Lipschitz neighborhood retraction to push this homotopy into $N$.


