The set of critical values of a sufficiently smooth function between manifolds has measure zero, meaning almost every value is a regular value. Proved by Arthur Sard, it is a foundational tool of differential topology, underlying transversality arguments and degree theory.
Facts
StatementThe set of critical values of a sufficiently smooth function between Euclidean spaces or manifolds has Lebesgue measure zero, so almost every point in the target is a regular value. 1 Proof YearThe one-dimensional special case (target dimension m=1) was proven by Anthony P. Morse in 1939; Arthur Sard proved the general case in 1942. Classification
Statement FormCharacterization Theorem 1 Connections
Has Statement Form
In Branch
Principal Result Of
Source Anthony Morse (Wikipedia)
Source Arthur Sard (Wikipedia)
Proved By
Source Sard's Theorem (Wikipedia)
Sources
1. Sard's Theorem (Wikipedia)
Wikimedia FoundationStatement section
...was proven by Anthony P. Morse in 1939, and the general case by Arthur Sard in 1942.
In Group: Geometric and Differential Topology, Lead
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
Proved By: Arthur Sard, Lead sentence
The theorem is named for Anthony Morse and Arthur Sard.
View the Source Arthur Sard (Wikipedia)
Principal Result Of: Arthur Sard, Lead, Sards theoremQuote, Principal Result Of: Arthur Sard, Lead, Sards theorem
His fame stems primarily from Sard's theorem, which says that the set of critical values of a differential function which has sufficiently many derivatives has measure zero.
Anthony Morse (Wikipedia)
Principal Result Of: Anthony Morse, LeadQuote, Principal Result Of: Anthony Morse, Lead
He is also known for his work on the Morse-Sard theorem and the Federer-Morse theorem.
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