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.
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Statement
More explicitly (Sternberg (1964, Theorem II.3.1); Sard (1942)), let
be
Intuitively speaking, this means that although
More generally, the result also holds for mappings between second countable differentiable manifolds
consists of those points at which the differential
has rank less than
Variants
There are many variants of this lemma, which plays a basic role in singularity theory among other fields. The case
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