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In mathematics, **Sard's theorem**, also known as **Sard's lemma** or the **Morse–Sard theorem**, is a result in mathematical analysis that asserts that the set of critical values (that is, the image of the set of critical points) of a smooth function *f* from one Euclidean space or manifold to another is a null set, i.e., it has Lebesgue measure 0. This makes the set of critical values "small" in the sense of a generic property. The theorem is named for Anthony Morse and Arthur Sard.

More explicitly,^{[1]} let

be , (that is, times continuously differentiable), where . Let denote the *critical set* of which is the set of points at which the Jacobian matrix of has rank . Then the image has Lebesgue measure 0 in .

Intuitively speaking, this means that although may be large, its image must be small in the sense of Lebesgue measure: while may have many critical *points* in the domain , it must have few critical *values* in the image .

More generally, the result also holds for mappings between differentiable manifolds and of dimensions and , respectively. The critical set of a function

consists of those points at which the differential

has rank less than as a linear transformation. If , then Sard's theorem asserts that the image of has measure zero as a subset of . This formulation of the result follows from the version for Euclidean spaces by taking a countable set of coordinate patches. The conclusion of the theorem is a local statement, since a countable union of sets of measure zero is a set of measure zero, and the property of a subset of a coordinate patch having zero measure is invariant under diffeomorphism.

There are many variants of this lemma, which plays a basic role in singularity theory among other fields. The case was proven by Anthony P. Morse in 1939,^{[2]} and the general case by Arthur Sard in 1942.^{[1]}

A version for infinite-dimensional Banach manifolds was proven by Stephen Smale.^{[3]}

The statement is quite powerful, and the proof involves analysis. In topology it is often quoted — as in the Brouwer fixed-point theorem and some applications in Morse theory — in order to prove the weaker corollary that “a non-constant smooth map has **at least one** regular value”.

In 1965 Sard further generalized his theorem to state that if is for and if is the set of points such that has rank strictly less than , then the *r*-dimensional Hausdorff measure of is zero.^{[4]} In particular the Hausdorff dimension of is at most *r*. Caveat: The Hausdorff dimension of can be arbitrarily close to *r*.^{[5]}

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^{a}^{b}Sard, Arthur (1942), "The measure of the critical values of differentiable maps",*Bulletin of the American Mathematical Society*,**48**(12): 883–890, doi:10.1090/S0002-9904-1942-07811-6, MR 0007523, Zbl 0063.06720. **^**Morse, Anthony P. (January 1939), "The behaviour of a function on its critical set",*Annals of Mathematics*,**40**(1): 62–70, doi:10.2307/1968544, JSTOR 1968544, MR 1503449.**^**Smale, Stephen (1965), "An Infinite Dimensional Version of Sard's Theorem",*American Journal of Mathematics*,**87**(4): 861–866, doi:10.2307/2373250, JSTOR 2373250, MR 0185604, Zbl 0143.35301.**^**Sard, Arthur (1965), "Hausdorff Measure of Critical Images on Banach Manifolds",*American Journal of Mathematics*,**87**(1): 158–174, doi:10.2307/2373229, JSTOR 2373229, MR 0173748, Zbl 0137.42501 and also Sard, Arthur (1965), "Errata to*Hausdorff measures of critical images on Banach manifolds*",*American Journal of Mathematics*,**87**(3): 158–174, doi:10.2307/2373229, JSTOR 2373074, MR 0180649, Zbl 0137.42501.**^**"Show that`f(C)`has Hausdorff dimension at most zero",*Stack Exchange*, July 18, 2013

- Hirsch, Morris W. (1976),
*Differential Topology*, New York: Springer, pp. 67–84, ISBN 0-387-90148-5. - Sternberg, Shlomo (1964),
*Lectures on Differential Geometry*, Englewood Cliffs, NJ: Prentice-Hall, MR 0193578, Zbl 0129.13102.