Does there exist a diffeomorphism $\varphi: \Bbb R^n\to \Bbb R^n$ such that $\varphi|_U=f$?

Question

Suppose $U,V$ are open subsets of $\Bbb R^n$ and $f: U\to V$ is a diffeomorphism, does there exist a diffeomorphism $\varphi: \Bbb R^n\to \Bbb R^n$ such that $\varphi|_U=f$?

Answer 1

No, not in general. Pick $n=1$, $U=(0;1)$, $V=(1;\infty)$ and $f(x)=1/x$. In fact you cannot even extend this to a continuous map on the whole space.