Products of Baire one double star functions
type: Item
instance of: scholarly article
title: Products of Baire one double star functions
zbMATH Open document ID: 1248.26006
author: Agnieszka Lukasiewicz
author: Aleksander Maliszewski
published in: Real Analysis Exchange
publication date: 2012-05-10T00:00:00Z
Mathematics Subject Classification ID: 26A21
Mathematics Subject Classification ID: 26A15
zbMATH DE Number: 6032451
zbMATH Keywords: Baire one star function
zbMATH Keywords: Baire one double star function
zbMATH Keywords: product of functions
reviewed by: Szymon Glab
MaRDI profile type: Publication
full work available at URL: realanalexch.36.2.0269
OpenAlex ID: W1498028716
DOI: 10.14321/REALANALEXCH.36.2.0269
review text: The symbol \(\mathcal C(f)\) stands for the set of points of continuity of a function \(f\). Define \([f=a]=\{x\in\mathbb{R}:f(x)=a\}\); the symbols \([f\neq a]\), \([f>a]\), etc. are defined analogously. A function \(f:\mathbb{R}\to\mathbb{R}\) is called a Baire one star function, if for each nonempty closed set \(P\subset\mathbb{R}\), there is a nonempty portion \(Q=P\cap(a,b)\) of \(P\) such that \(f|_Q\) is continuous. For a given interval \(I\subset\mathbb{R}\) and a function \(\psi:I\to\mathbb{R}\) define by induction on \(\alpha<\omega_1\) NEWLINE\[NEWLINE \mathcal U_\alpha(\psi)=\text{int}\Big(\bigcup_{\beta<\alpha}\mathcal U_\alpha(\psi)\cup\mathcal C\big(\psi|_{I\setminus\bigcup_{\beta<\alpha}\mathcal U_\alpha(\psi)}\big)\Big). NEWLINE\]NEWLINE Let \(\mathcal S_\alpha=\{f:\mathbb{R}\to\mathbb{R}:\mathcal U_\alpha(f)=\mathbb{R}\}\). In particular, \(\mathcal S_0\) is the class of all continuous functions and \(\mathcal S_1\) is the class \(\mathcal B_1^{**}\).NEWLINENEWLINE\textit{A. Lukasiewicz} proved [``A classification of Baire one star functions in topological spaces'', Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 54, No. 1--2, 175--181 (2006; Zbl 1387.54011)] that \(\{\mathcal S_\alpha:\alpha<\omega_1\}\) is a classification of Baire one star functions.NEWLINENEWLINEThe authors' goal is to make the first step toward the characterization of the products of functions from these classes. The main result of the paper is the following.NEWLINENEWLINETheorem. Let \(f:\mathbb{R}\to\mathbb{R}\). Denote by \(\mathcal I\) the family of all bounded connected components \(I=(a,b)\) of \(\mathcal U_0(f)\) with the property that \(f(a)f(b)<0\) and \(I\cap[f=0]=\emptyset\). The following are equivalent:NEWLINENEWLINE(i) there exists \(g,h\in\mathcal S_1\) such that \(f=gh\) on \(\mathbb{R}\);NEWLINENEWLINE(ii) \(f\in\mathcal S_2\) and for each \(x\in[f\neq 0]\setminus\mathcal U_1(f)\): NEWLINE\[NEWLINE \exists\delta>0\;\;\;(x-\delta,x+\delta)\cap\big(\bigcup_{I\i…
description: scientific article; zbMATH DE number 6032451