Critical values lie on a line (Q1773551)

The main result of the article under review is the following theorem. Let \(F\colon {\mathbb R}^n \rightarrow {\mathbb R}^m\) be a function whose all the partial derivatives of order \(k\) satisfy the Hölder condition with exponent \(\lambda \in [0,1).\) Let \(C_p(F)= \cup_{r=0}^p\{x\in {\mathbb R}^n\colon \text{rank} (DF(x))=r\}\) be the \(p\)-critical set of \(F.\) Set \(\mu = \text{max}\{1/(p+((n-p)/(k+\lambda))), 1/m\}.\) Then there is a function \(f\colon {\mathbb R} \rightarrow {\mathbb R}^m\) of smoothness class \(<\mu,\) that is, \(f\in C^k\) for every \(k<\mu,\) such that \(F(C_p(F))\subseteq f(\Sigma_\mu f).\) Here the set \(\Sigma_\mu f\) consists of all the points \(x\in {\mathbb R}\) such that any partial derivatives of \(f\) of order \(<\mu\) vanishes at \(x.\) The proof is mainly based on an idea of Peano [\textit{H. Sagan}, Space-filling curves. Universitext (New York: Springer-Verlag) (1994; Zbl 0806.01019)]. The author remarks that his result may be considered as a Sard-type theorem and the sharpness of the value \(\mu\) follows from earlier results, where necessary and sufficient conditions for the Morse-Sard theorem are established for some particular cases (see, for example [\textit{S. M. Bates} and \textit{A. Norton}, Duke Math. J. 83, No.~2, 399--413 (1996; Zbl 0877.58008)]).