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1997/1998 Characterizing Derivatives by Preimages of Sets
Krzysztof Ciesielski
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Real Anal. Exchange 23(2): 553-566 (1997/1998).


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 \(\mathcal{D}=\{g^{-1}(A)\colon A\in\mathcal{A}\ \&\ g\in\Delta\}\).


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Krzysztof Ciesielski. "Characterizing Derivatives by Preimages of Sets." Real Anal. Exchange 23 (2) 553 - 566, 1997/1998.


Published: 1997/1998
First available in Project Euclid: 14 May 2012

zbMATH: 0943.26015
MathSciNet: MR1639976

Primary: 26A24
Secondary: 03E35.

Keywords: {derivatives} , {preimages of sets.}

Rights: Copyright © 1999 Michigan State University Press

Vol.23 • No. 2 • 1997/1998
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