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manuscripta mathematica

, Volume 129, Issue 3, pp 401–408 | Cite as

The curve selection lemma and the Morse–Sard theorem

  • Carlos Gustavo MoreiraEmail author
  • Maria Aparecida Soares Ruas
Article

Abstract

We use an inequality due to Bochnak and Lojasiewicz, which follows from the Curve Selection Lemma of real algebraic geometry in order to prove that, given a C r function \({f\colon U\subset{{\mathbb R}^m}\to{\mathbb R}}\), we have
$$\lim\limits_{\substack{y\to x \\ y\in \text{crit}(f)}} \frac{|f(y)-f(x)|}{|y-x|^r}=0, \hbox{for all} \;x\in \text{crit}(f)' \cap U,$$
where \({\text{crit}(f)= \{x\in U \mid df(x)=0\}}\). This shows that the so-called Morse decomposition of the critical set, used in the classical proof of the Morse–Sard theorem, is not necessary: the conclusion of the Morse decomposition lemma holds for the whole critical set. We use this result to give a simple proof of the classical Morse–Sard theorem (with sharp differentiability assumptions).

Mathematics Subject Classification (2000)

Primary 58K05 Secondary 32S05 

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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Carlos Gustavo Moreira
    • 1
    Email author
  • Maria Aparecida Soares Ruas
    • 2
  1. 1.Instituto de Matemática Pura e Aplicada (IMPA)Rio de JaneiroBrazil
  2. 2.Departamento de Matemática, Instituto de Ciências Matemáticas e de ComputaçãoUniversidade de São PauloSão CarlosBrazil

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