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The set of the critical values of a smooth function has measure zero
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
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.
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 is for and if is the set of points such that has rank strictly less than , then the r-dimensional Hausdorff measure of is zero. In particular the Hausdorff dimension of is at most r. Caveat: The Hausdorff dimension of can be arbitrarily close to r.