Does a diffeomorphism between the interiors of two manifolds extend to the manifolds?

Question

Let $M,N$ be manifolds with boundary. Let $f:\mathring{M}\to \mathring{N}$ be a diffeomorphism of their interiors. Does it extend to a diffeomorphism $M\to N$?

I suppose we can look at the problem locally, and then the problem is: if $U,V\subset \mathbb{H}^n$ are open sets in half-space, does a diffeomorphism $f:\mathring{U}\to \mathring{V}$ extend to a diffeomorphism $f:U\to V$?

EDIT: Seeing as the answer is negative, I now ask: what are sufficient conditions for the assertion to be true?

Answer 1

Take the inversion map on $(0, \infty)$ given by $f(x) = \frac{1}{x}$. This is differentiable with differentiable inverse but you cannot extend this to find $f(0)$

Answer 2

Consider $M = N = [0, \infty)$ and $$f: (0, \infty) \longrightarrow (0, \infty),$$ $$x \mapsto \frac{1}{x}.$$ $f$ is a diffeomorphism which doesn't even have a continuous extension to all of $M = [0, \infty)$.