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## Summary

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.

## 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 $X\subset \mathbb {R} ^{n}$  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.