The curve selection lemma and the Morse-Sard theorem (Q2391152)

The authors prove that if \(U\) is an open set in \({\mathbb R}^n\), \(f:U \to\mathbb R\) is a \(C^r\)-function, \(C(f) \subset U\) is the set of critical points of \(f\) and \(x\) an accumulation point of \(C(f)\), then: \[ \lim _{y \to x} \frac{f(y)-f(x)}{y-x}=0, \] where \(y\) varies in \(C(f)\). This is shown using an inequality due to \textit{J. Bochnak} and \textit{S. Łojasiewicz} [Proc. Liverpool Singularities-Sympos. I, Dept. Pure Math. Univ. Liverpool 1969--1970, 254--261 (1971; Zbl 0221.58002)], whose proof (included in the present paper) involves Milnor's curve selection lemma. They use the limit above to give a simplified proof of the Morse-Sard's theorem, which says that the set of critical values of a \(C^{m-n+1}\)-function \(g:U \to {\mathbb R}^n\), where \(U\) is an open subset of \({\mathbb R}^m\), is a subset of \({\mathbb R}^n\) of measure zero. The usual proof involves a partition of \(U\) by means of subsets \(A_j\) with special properties (the so-called Morse decomposition), and a more complicated induction. They give examples showing that the conclusion of Morse-Sard's theorem is not necessarily valid for functions of class \(C^r\), if \(r < m+n-1\).