Characterizing Derivatives by Preimages of Sets

Abstract

In this note we will show that many classes \(\mathcal{F}\) of real functions \(f\colon {\mathbb R}\to\mathbb{R}\) can be characterized by preimages of sets in a sense that there exist families \(\mathcal{A}\) and \(\mathcal{D}\) of subsets of \(\mathbb{R}\) such that \(\mathcal{F}=\mathcal{C}(\mathcal{D},\mathcal{A})\), where \(\mathcal{C}(\mathcal{D},\mathcal{A})=\{f\in\mathbb{R}^\mathbb{R}\colon f^{-1}(A)\in \mathcal{D}\ \text{ for every } A\in\mathcal{A}\}.\) In particular, we will show that there exists a Bernstein \(B\subset \mathbb{R}\) such that the family \(\Delta\) of all derivatives can be represented as \(\Delta=\mathcal{C}(\mathcal{D},\mathcal{A})\), where \(\mathcal{A}=\bigcup_{c\in\mathbb{R}}\{(-\infty,c),(c,\infty),B+c\}\) and \(\mathcal{D}=\{g^{-1}(A)\colon A\in\mathcal{A}\ \&\ g\in\Delta\}\).