## Real Analysis Exchange

### Characterizing Derivatives by Preimages of Sets

Krzysztof Ciesielski

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

#### Article information

Source
Real Anal. Exchange, Volume 23, Number 2 (1999), 553-566.

Dates
First available in Project Euclid: 14 May 2012

https://projecteuclid.org/euclid.rae/1337001365

Mathematical Reviews number (MathSciNet)
MR1639976

Zentralblatt MATH identifier
0943.26015

#### Citation

Ciesielski, Krzysztof. Characterizing Derivatives by Preimages of Sets. Real Anal. Exchange 23 (1999), no. 2, 553--566. https://projecteuclid.org/euclid.rae/1337001365