Products of Darboux functions
type: Item
instance of: scholarly article
title: Products of Darboux functions
zbMATH Open document ID: 0770.26005
author: Tomasz Natkaniec
published in: Real Analysis Exchange
publication date: 1993-06-29T00:00:00Z
review text: In this paper, there are three theorems about the existence of a function \(g: I\to\mathbb{R}\) for a given set \(F\) of real functions defined on the closed interval \(I\) such that \(fg\) is Darboux for any \(f\in F\). Theorem 1. For any countable set \(F\) of Baire functions of the class \(B_ \alpha\), where \(\alpha>1\), there exists a non-zero Darboux function \(g\) in \(B_ \alpha\) such that: (1) \(fg\) is Darboux for any \(f\in F\) and (2) the set \(\{x\in I: g(x)\neq 0\}\) is of the first Baire category in \(\mathbb{R}\) and of Lebesgue measure zero. Theorems 2 and 3 are proved under the hypothesis that the following set- theoretical assertion is true: Any union of less than \(2^ \omega\) sets of the first category in \(\mathbb{R}\) is also of the first category in \(\mathbb{R}\).
reviewed by: Ladislav Mišík
Mathematics Subject Classification ID: 26A15
zbMATH DE Number: 222102
zbMATH Keywords: Darboux property
zbMATH Keywords: Baire functions of the class \(\alpha\)
zbMATH Keywords: Darboux function
zbMATH Keywords: first Baire category
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 222102