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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A note on a corollary of Sard's theorem


Author: John C. Wells
Journal: Proc. Amer. Math. Soc. 48 (1975), 513-514
MSC: Primary 26A63; Secondary 28A75
MathSciNet review: 0364578
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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\vert{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.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0364578-4
Article copyright: © Copyright 1975 American Mathematical Society