Sard theorem

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2020 Mathematics Subject Classification: Primary: 58A05 Secondary: 28A [MSN][ZBL]

Theorem Let $M$ and $N$ be two $C^r$ manifolds and $f:M\to N$ a $C^r$ map. If $r> \max \{0, \dim M - \dim N\}$, then the critical values of $f$ form a set of measure zero. Therefore the set of regular values (see Singularities of differentiable mappings) has full measure.

The theorem was proved by A. Sard in [Sa]. Observe that there is no uniquely defined measure on $N$ and the statement means that, if $S\subset N$ denotes the (closed) subset of singular values of $f$, then, for every chart $(U, \phi)$ in the atlas defining $N$, $\phi (U\cap S)$ is a set of (Lebesgue) measure zero.

As a corollary of Sard's theorem we conclude that the set of regular values is dense. Thus $S$ is a meager set. The latter statement is also sometimes called Sard's theorem: however it is not equivalent to the one above, since closed meager sets might have positive Lebesgue measure.


[Sa] A. Sard, "The measure of critical values of differentiable maps" Bull. Amer. Math. Soc. , 48 (1942) pp. 883–890 MR7523 Zbl 0063.06720
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Sard theorem. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article