# Sard theorem

From Encyclopedia of Mathematics

Let be a -mapping of manifolds and of dimensions and , respectively; if , then the critical values (cf. Critical value) of form a set of measure zero. The set of regular values turns out to be of full measure and everywhere dense. The theorem was proved by A. Sard [1].

#### References

[1] | A. Sard, "The measure of critical values of differentiable maps" Bull. Amer. Math. Soc. , 48 (1942) pp. 883–890 |

#### Comments

"Full measure" is, in the Russian article, called "massive setmassive" . See also Singularities of differentiable mappings.

**How to Cite This Entry:**

Sard theorem.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Sard_theorem&oldid=11998

This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article