# Sard's theorem

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 (Sternberg (1964, Theorem II.3.1); Sard (1942)), let

${\displaystyle f\colon \mathbb {R} ^{n}\rightarrow \mathbb {R} ^{m}}$

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

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

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

${\displaystyle f:N\rightarrow M}$

consists of those points at which the differential

${\displaystyle df:TN\rightarrow TM}$

has rank less than ${\displaystyle m}$ as a linear transformation. If ${\displaystyle k\geq \max\{n-m+1,1\}}$, then Sard's theorem asserts that the image of ${\displaystyle X}$ has measure zero as a subset of ${\displaystyle M}$. 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 ${\displaystyle m=1}$ was proven by Anthony P. Morse in 1939 (Morse 1939), and the general case by Arthur Sard in 1942 (Sard 1942).

A version for infinite-dimensional Banach manifolds was proven by Stephen Smale (Smale 1965).

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 use the weaker corollary that “a non-constant smooth map has a regular value”.

In 1965 Sard further generalized his theorem to state that if ${\displaystyle f:M\rightarrow N}$ is ${\displaystyle C^{k}}$ for ${\displaystyle k\geq \max\{n-m+1,1\}}$ and if ${\displaystyle A_{r}\subseteq M}$ is the set of points ${\displaystyle x\in M}$ such that ${\displaystyle df_{x}}$ has rank strictly less than ${\displaystyle r}$, then the r-dimensional Hausdorff measure of ${\displaystyle f(A_{r})}$ is zero. In particular the Hausdorff dimension of ${\displaystyle f(A_{r})}$ is at most r. Caveat: The Hausdorff dimension of ${\displaystyle f(A_{r})}$ can be arbitrarly close to r.[1]