Sard's theorem

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 $$<m$$. 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.