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
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).
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Moreira, C.G., Ruas, M.A.S. The curve selection lemma and the Morse–Sard theorem. manuscripta math. 129, 401–408 (2009). https://doi.org/10.1007/s00229-009-0275-2
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DOI: https://doi.org/10.1007/s00229-009-0275-2