Let $M,N$ be smooth manifolds with boundary (of the same dimension). Let $f:M \to N$ be a smooth map satisfying
$(1) \, \,f(\partial M)=\partial N,f(\operatorname{Int} M)=\operatorname{Int} N$.
$(2) \, \, df_p$ is invertible for every $p \in M$.
Is it true that $f$ is a local homeomorphism?
I suspect $f$ must in fact be a local diffeomorphism**:
It is certainly a local diffeomorphism around each point in $\operatorname{Int} M$ (By the inverse function theorem).
The question is what happens at the boundary $\partial M$:
Clearly $f|_{\partial M}:\partial M \to \partial N$ is a local diffeomorphism (again by the inverse function for manifolds without boundary).
However, this does not immediately imply that $f$ (when considered as a map $M \to N$) is a local diffeomorphism.
** To say that $f\colon M\to N$ is a local diffeomorphism means that each point of $M$ has an open neighborhood $U$ such that $f(U)$ is open in $N$ and $f|_U$ is a diffeomorphism from $U$ onto $f(U)$.