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
Sources
1. Sard's Theorem (Wikipedia)
Wikimedia FoundationStatement sectionQuote, Statement section
...was proven by Anthony P. Morse in 1939, and the general case by Arthur Sard in 1942.
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