fibers of a local diffeomorphism discrete set

Question

Let $M$ and $N$ be smooth manifolds with or without boundary and $f : M \to N$ be a local diffeomorphism. I have to show that for each point $q$ of $N$, $f^{-1}(q)$ is a discrete subset of $M$. But I am just stuck. The only hint I have is that each point $f^{-1}(q)$ has a neighborhood on which $f$ is injective

Could anyone please help me?

Answer 1

In general, the preimage theorem for maps $f:M \to N$, with dimension $m,n$ respectively and $m>n$, and $q$ a regular value, then $f^{-1}(q)$ is a submanifold of dimension $n-k$. Hence, for a local diffeomorphism, $q$ should be a regular value (locally injective) and $M,N$ have the same dimension.