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


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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