False statementConstruction: Literature-sourcedVerification: Not suppliedaudited 2026-09-01

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Sard's theorem does not hold for every $C^{1}$ map

Statement

False claim: Sard's theorem remains true for every $C^{1}$ map between arbitrary dimensions.

The differentiability threshold in Morse-Sard for Euclidean maps is real. Standard source treatments record Whitney's construction of a $C^{1}$ map $R^{2} \to R$ whose critical values contain a set of positive measure, so the naive $C^{1}$ form of Sard fails once the hypothesis $r > \max {m - n, 0}$ is dropped.

Depends on

Morse-Sard for Euclidean maps

Used by

A C¹ map whose critical values have positive measure Counterexample

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Sources

Encyclopedia of Mathematics, Sard theorem (standard reference, not scraped)
John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)

Sard's theorem does not hold for every C1C1 map