Let $(X,d)$ be a metric space where $X$ is a manifold and $\widetilde{X}$ its metric completion. Further let $X\xrightarrow{f}Y$ be a local diffeomorphism.
Can $f$ be extended to $\widetilde{X}\xrightarrow{\widetilde{f}}Y$, and if so do we have fruther information (e.g. uniqueness, structure)?
Edit
In my case, $Y$ is a Riemannian manifold $(Y,g)$ and the distance $d$ on $X$ is the one induced by the pullback metric $f^*g$.
It appears that $f$ can be extended if and only if $f$ is Cauchy-continuous. I believe this should give me the existence and uniqueness of the extension.