Some typical results on bounded Baire 1 functions
type: Item
instance of: scholarly article
title: Some typical results on bounded Baire 1 functions
zbMATH Open document ID: 0542.26004
DOI: 10.1007/BF01958029
published in: Acta Mathematica Hungarica
publication date: 1984-01-01T00:00:00Z
review text: In the paper the authors deal with functions f:[0,1]\(\to R\) (R - the real line). Let b\({\mathcal A}\), \(b\Delta\), b\({\mathcal D}{\mathcal B}^ 1\), b\({\mathcal B}^ 1\) be Banach spaces of bounded approximately continuous functions, bounded derivatives, bounded Daboux Baire 1 functions and bounded Baire 1 functions, resp., with norm \(\| f\| =\sup | f|.\) Some typical properties of functions of the above spaces are given. These concern of level sets, of continuity points and of the range. Let us introduce some of the reached results: Theorem 1.10. The set of functions f such that \(f^{-1}(y)\) is a nowhere dense (Lebesgue) nullset for all \(y\in R\) is a residual \(G_{\delta}\) set in any of b\({\mathcal A}\), \(b\Delta\) and b\({\mathcal D}{\mathcal B}^ 1\). Theorem 2.4. Let \(\mu\) be an arbitrary finite Borel measure on [0,1] and \({\mathcal F}=b{\mathcal A},\quad b\Delta,\quad b{\mathcal D}{\mathcal B}^ 1,\quad b{\mathcal B}^ 1.\) Then \(\{f\in {\mathcal F}:\mu(C_ f)=0\}\) is an everywhere dense \(G_{\delta}\) set in \({\mathcal F} (C_ f\)- the set of continuity points of f). Theorem 3.5. Let \({\mathcal F}=b{\mathcal A},\quad b\Delta,\quad b{\mathcal D}{\mathcal B}^ 1,\quad b{\mathcal B}^ 1.\) Then the family \(\{f\in {\mathcal F}: f(C_ f)\) is of power of continuum\} is an everywhere dense \(G_{\delta}\) set in \({\mathcal F}\).
Mathematics Subject Classification ID: 26A21
Mathematics Subject Classification ID: 26A15
Mathematics Subject Classification ID: 26A24
zbMATH DE Number: 3861477
zbMATH Keywords: Banach spaces
zbMATH Keywords: bounded approximately continuous functions
zbMATH Keywords: bounded derivatives
zbMATH Keywords: bounded Daboux Baire 1 functions
zbMATH Keywords: bounded Baire 1 functions
zbMATH Keywords: level sets
zbMATH Keywords: range
zbMATH Keywords: finite Borel measure
zbMATH Keywords: set of continuity points
author: Andrew M. Bruckner
author: Gy. Petruska
reviewed by: Pavel Kostyrko
MaRDI profile type: Publication
cites work: Differentiation of real functions
cites work: Most Darboux Baire 1 functions map big sets onto small sets
cites work: Sur La Premiere Derivee
full work available at URL: https://doi.org/10.1007/bf01958029
OpenAlex ID: W2067177589
description: scientific article; zbMATH DE number 3861477