On sets of critical values in the real line (Q1918381)

Let \({\mathcal C}\) be a class of functions in \(C^1(\mathbb{R}, \mathbb{R})\) and let denote by \(\text{CV} ({\mathcal C})\) the collection of all sets \(B\) of the form \(B=\text{CV}(f)\) for some \(f\in {\mathcal C}\) with compactly supported derivatives, where \(\text{CV}(f)\) is the set of critical values of \(f\). The main purpose of this interesting paper is to characterize the elements of \(CV({\mathcal C})\) for some particular smoothness class: \({\mathcal C}= C^s(\mathbb{R}, \mathbb{R})\), \(s>1\), \({\mathcal C}= C^\infty (\mathbb{R}, \mathbb{R})\), etc.