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 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, let be , (that is, times continuously differentiable), where . Let denote the critical set of which is the set of points at which the Jacobian matrix of has rank . Then the has Lebesgue measure 0 in . Intuitively speaking, this means that although may be large, its image must be small in the sense of Lebesgue measure: while may have many critical points in the domain , it must have few critical values in the image . More generally, the result also holds for mappings between differentiable manifolds and of dimensions and , respectively. The critical set of a function consists of those points at which the differential has rank less than as a linear transformation. If , then Sard's theorem asserts that the image of has measure zero as a subset of . 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 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.