The curve selection lemma and the Morse–Sard theorem

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).