Suppose $M$ and $N$ are smooth manifolds of dimension $m$ and $n$ respectively and $f\colon M\to N$ is a $\mathrm{C}^r$-smooth map, where $r\ge1$ and $r\gt m-n$. Then the set of critical values in $N$ is a meager subset (alias first category subset) and a negligible subset (alias measure zero subset) of $N$. In particular, the set of regular values is dense in $N$. Furthermore, the $f$-image of points of $M$ where $f$ has rank at most $r$ ($0\lt r\lt m$) has Hausdorff dimension at most $r$.

If $N$ is a Banach manifold and $q\ge1$, $f$ is a Fredholm map, and $q$ is strictly greater than the index of $f$, then the critical values of $f$ form a meager subset of $N$.