Sard's theorem

Sard's theorem states that the critical values of a sufficiently smooth map between finite-dimensional manifolds have measure zero. It places no restriction on the size or shape of the critical-point set itself.

Critical point: A point where the derivative of a smooth map fails to have maximal rank. Sard's theorem begins by collecting the images of these points.
Smooth map: A map between smooth manifolds whose coordinate representations have derivatives of every order. Sard's theorem applies to this class of maps, with finite differentiability sufficient in fixed dimensions.
Arthur Sard: American mathematician who proved a general theorem on critical values of smooth maps. His 1942 paper established the result now bearing his name.
Morse theory: A method relating a manifold's topology to the critical points of a smooth function. Sard's theorem supplies regular levels between critical values.
Sard's theorem for Banach spaces: Extensions of Sard's theorem to maps between infinite-dimensional Banach spaces under additional hypotheses. The finite-dimensional theorem does not automatically extend to infinite-dimensional settings.
Critical value: The image of a critical point under a smooth map. The theorem's measure-zero conclusion concerns precisely these values.
Rank of a linear map: The dimension of the image of a linear transformation. A derivative has deficient rank exactly at a critical point.
Anthony Morse: American mathematician whose work on critical points and measure helped shape early versions of Sard's theorem. Morse proved an earlier result that Sard later strengthened.
Morse function: A smooth real-valued function whose critical points all have nonsingular Hessians. Sard's theorem helps select regular values used to study its level sets.
Sard's theorem for finite differentiability: Versions of Sard's theorem specifying how many continuous derivatives suffice for given dimensions. The smooth hypothesis can be weakened to a dimension-dependent differentiability threshold.

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