Sard's theorem

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Sard's theorem, also known as Sard's lemma or the Morse–Sard theorem, is a result in mathematical analysis which 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 (Sternberg (1964, Theorem II.3.1); Sard (1942)), 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 < 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 second countable 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.


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 (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 is involved 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”, and sometimes “...hence also a regular point”.

In 1965 Sard further generalized his theorem to state that if f:M\rightarrow N is C^k for k\geq \max\{n-m+1, 1\} and if A_r\subseteq M is the set of points x\in M such that df_x has rank less than or equal to r, then f(A_r) has Hausdorff dimension at most r.

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