# Sard's theorem

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In mathematics, Sard's theorem, also known as Sard's lemma or the MorseSard 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.

## Statement

More explicitly,  let

$f\colon \mathbb {R} ^{n}\rightarrow \mathbb {R} ^{m}$ be $C^{k}$ , (that is, $k$ times continuously differentiable), where $k\geq \max\{n-m+1,1\}$ . Let $X$ denote the critical set of $f,$ which is the set of points $x\in \mathbb {R} ^{n}$ at which the Jacobian matrix of $f$ has rank $ . Then the image $f(X)$ has Lebesgue measure 0 in $\mathbb {R} ^{m}$ .

Intuitively speaking, this means that although $X$ may be large, its image must be small in the sense of Lebesgue measure: while $f$ may have many critical points in the domain $\mathbb {R} ^{n}$ , it must have few critical values in the image $\mathbb {R} ^{m}$ .

More generally, the result also holds for mappings between differentiable manifolds $M$ and $N$ of dimensions $m$ and $n$ , respectively. The critical set $X$ of a $C^{k}$ function

$f:N\rightarrow M$ consists of those points at which the differential

$df:TN\rightarrow TM$ has rank less than $m$ as a linear transformation. If $k\geq \max\{n-m+1,1\}$ , then Sard's theorem asserts that the image of $X$ has measure zero as a subset of $M$ . 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.

## Variants

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

A version for infinite-dimensional Banach manifolds was proven by Stephen Smale. 

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 $f:N\rightarrow M$ is $C^{k}$ for $k\geq \max\{n-m+1,1\}$ and if $A_{r}\subseteq N$ is the set of points $x\in N$ such that $df_{x}$ has rank strictly less than $r$ , then the r-dimensional Hausdorff measure of $f(A_{r})$ is zero.  In particular the Hausdorff dimension of $f(A_{r})$ is at most r. Caveat: The Hausdorff dimension of $f(A_{r})$ can be arbitrarily close to r. 

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1. Sard, Arthur (1942), "The measure of the critical values of differentiable maps", Bulletin of the American Mathematical Society , 48 (12): 883–890, doi:, MR   0007523, Zbl   0063.06720.
2. 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.
3. 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.
4. 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.
5. "Show that f(C) has Hausdorff dimension at most zero", Stack Exchange , July 18, 2013