Sierpiński-Zygmund uniform limits of extendable connectivity functions.
type: Item
instance of: scholarly article
title: Sierpiński-Zygmund uniform limits of extendable connectivity functions.
zbMATH Open document ID: 1048.26005
author: Sumit K. Garg
published in: Real Analysis Exchange
publication date: 2004-08-06T00:00:00Z
review text: \(f:{\mathbb R}\to{\mathbb R}\) is: a Sierpiński-Zygmund (SZ) function if \(f| A\) is continuous for no subset \(A\) of \(\mathbb R\) of cardinality of the continuum; an extendable (Ext) function if there is a function \(F:{\mathbb R}\times [0,1]\to{\mathbb R}\) such that \(F(x,0)=f(x)\) for all \(x\in{\mathbb R}\) and \(F| C\) is connected for each connected subset \(C\subset{\mathbb R}\times [0,1]\). It is known that Ext\(\cap \text{SZ}=\emptyset\) [see e.g.,\textit{K. Banaszewski} and \textit{T. Natkaniec}, Real Anal. Exch. 24, No. 2, 827--835 (1998; Zbl 0967.26002)]. The first part of the paper under review contains the construction of a function \(f\colon{\mathbb R}\to{\mathbb R}\) which is the uniform limit of a sequence of extendable functions (i.e., \(f\in\overline{\text{Ext}}\)). In the second part of the paper it is shown under MA (in fact, under the assumption that every set \(A\subset\mathbb R\) of size less than the continuum is meager) that the class SZ\(\cap \overline{\text{Ext}}\) cannot be characterized by preimages of sets, i.e., there are no \({\mathcal A},{\mathcal B}\subset{\mathcal P}(\mathbb R)\) such that SZ\(\cap \overline{ \text{Ext}}\) is equal to the class of all \(f\colon{\mathbb R}\to{\mathbb R}\) with \(f^{-1}(B)\in{\mathcal A}\) for each \(B\in {\mathcal B}\). Finally the author poses the question whether or not the class SZ\(\cap \overline{\text{Ext}}\) can be characterized by images of sets.
reviewed by: Tomasz Natkaniec
Mathematics Subject Classification ID: 26A15
Mathematics Subject Classification ID: 54C30
zbMATH DE Number: 2084023
zbMATH Keywords: Sierpiński-Zygmund function
zbMATH Keywords: extendable function
zbMATH Keywords: uniform limit
zbMATH Keywords: characterization by preimages
MaRDI profile type: Publication
full work available at URL: realanalexch.28.1.0105
OpenAlex ID: W1519985909
DOI: 10.14321/REALANALEXCH.28.1.0105
Recommended article: On Measurable Sierpiński-Zygmund Functions
Recommended article: Darboux like functions that are characterizable by images, preimages and associated sets
Recommended article: Q4227939
Recommended article: A symmetrically continuous function which is not countably continuous
Recommended article: Additive Sierpiński-Zygmund functions
Recommended article: On characterizing extendable connectivity functions by associated sets
Recommended article: A century of Sierpiński-Zygmund functions
Recommended article: More about Sierpiński-Zygmund uniform limits of extendable functions
Recommended article: An almost continuous nonextendable function
description: scientific article; zbMATH DE number 2084023