Topological properties of regular and critical points and values

Question

Let $f\colon M\rightarrow N$ be a smooth map between smooth manifolds.

Consider the following two statements, the second one under the assumption

1. The set of regular points of $f$ are open in $M$, the critical ones are closed in $M$.
2. The set of regular values of $f$ are open in $N$, the critical ones are closed in $N$.

I think the first statement holds in complete generality, the second one under the assumption that $f$ is a closed map, e.g. a proper one.

Am I correct?