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Named after Anthony P. Morse and Arthur Sard.
Recall the following definitions from diferential topology. The set of critical points of a smooth map $f$ is the set of points in the domain of $f$ where the tangent map is not surjective. The set of critical values of $f$ is the $f$-image of the set of critical points of $f$. The set of regular values of $f$ is the complement of the set of critical values of $f$.
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 $N$.
The case $n=1$:
The case $n\gt1$:
The case when $N$ is a Banach manifold:
The part concerning Hausdorff measures:
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