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)?**

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