Let $M,N$ be smooth manifolds of the same dimension, and suppose $M$ has a non-empty boundary. Let $f:M \to N$ be a smooth map, and suppose that $df_p$ is invertible for some $p \in \partial M$.
Is $f$ a local diffeomorphism onto its image around $p$?
That is, does there exist an open neighbourhood $U$ of $p$, such that $f(U)$ is a smooth submanifold with boundary of $N$, and $f|_U:U \to f(U)$ a diffeomorphism?
Is it even true that $f|_U$ must be injective for sufficiently small $U$ around $p$?
Note that 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 $f(U)$.
When $M$ has a non-empty boundary, a map with invertible differential needs not be open: Take e.g. $M = N=[0, \infty)$ and $f(x)=x+1$. However, in this case $f$ is a diffeomorphism onto its image.