A note on a corollary of Sard’s theorem
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- by John C. Wells PDF
- Proc. Amer. Math. Soc. 48 (1975), 513-514 Request permission
Abstract:
A corollary of Sard’s theorem is the following: Corollary. Let $f:K \to {R^n}$ be a smooth (i.e. $f \in {C^k},k \geq 1$) map defined on a compact subset $K$ of ${R^n}$. Let $C = \{ y|{f^{ - 1}}(y)\;is\;infinite\}$. Then the Lebesgue measure of $C$ is zero. The purpose of this note is to show that a similar version of this theorem holds for Lipschitz functions.References
- John W. Milnor, Topology from the differentiable viewpoint, University Press of Virginia, Charlottesville, Va., 1965. Based on notes by David W. Weaver. MR 0226651
- Arthur Sard, The measure of the critical values of differentiable maps, Bull. Amer. Math. Soc. 48 (1942), 883–890. MR 7523, DOI 10.1090/S0002-9904-1942-07811-6
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 48 (1975), 513-514
- MSC: Primary 26A63; Secondary 28A75
- DOI: https://doi.org/10.1090/S0002-9939-1975-0364578-4
- MathSciNet review: 0364578