I'm considering the so called "stack of records" theorem, which states that if $f: M \rightarrow N$ is a smooth function between manifolds of the same dimension, and $M$ is compact, then if $y \in N$ is a regular value of $f$, then $|f^{-1}(y)|$ is finite.
My question is, do we necessarily need $M$ to be boundaryless to conclude that $|f^{-1}(y)|$ is finite? The proofs I often see invoke the inverse function theorem, which to my knowledge doesn't necessarily hold if $M$ has boundary. However, it seems like the result should still go through...
One sufficient condition for this would be $|f^{-1}(y)| \subseteq int(M)$ for each regular value $y$ so that the implicit function theorem can be applied, but I was wondering if there was something more general that anybody knows of.
Thanks in advance!