I am reading Milnor's Topology from the differentiable viewpoint and I came across the following:
Let $f: M\rightarrow N$ be a smooth map between manifolds of the same dimension. Let $x \in M$ be a regular point.
It follows from inverse function theorem that $f$ maps a neighbourhood of $x$ in $M$ diffeomorphically onto an open set in $N$.
Here should we assume $x \in M$ and $y=f(x)\in N$ to not to be on the boundary to have an open neighbourhood around them in the first place?