How can we prove that $f^{-1}(q)$ is a closed subset in $M$ given $f:M\to N$ a smooth function between two manifolds and $q\in N$?

Question

I know that, since $\{q\}$ is a closed subset of $N$, $N\setminus\{q\}$ is an open subset of $N$. Since $f$ is smooth, in particular continuous, we could say that $f^{-1}(N\setminus\{q\})$ is an open subset of $M$, but this is just $M\setminus f^{-1}(q)$. So $f^{-1}(q)$ must be a closed subset in $M$.

Is this reasoning correct? I'm not sure if I can say right away that $f^{-1}(N\setminus\{q\})$ is also open. If not, how else would you prove this statement?