Some examples on continuous restrictions
type: Item
instance of: scholarly article
title: Some examples on continuous restrictions
zbMATH Open document ID: 0533.26002
author: Jack G. Ceder
published in: Real Analysis Exchange
publication date: 1982-01-01T00:00:00Z
review text: In the present paper the author raised the following question: Are there classes of functions, f: \(R\to R\) (R - the real line), not having the property of Baire, for which there exists a dense subset A of R with A uncountable such that the restriction \(f| A\) is continuous? Ingenious examples show that the Lebesgue measurable functions and the connected functions (i.e., the graph is a connected set), fail to have this property. A function f: \(R\to R\) is said to be countably decomposable into continuous functions if there exist countably many disjoint sets \(A_ n\), \(n=1,2,...\), whose union is R such that \(f| A_ n\) is continuous for each n. It is shown that there exists a bounded, approximately continuous, lower-semi-continuous function which is not countably decomposable into continuous functions. This result improves a result of \textit{R. O. Davies} [Čas. Pěstovani Mat. 98, 398-399 (1973; Zbl 0272.26003)].
Mathematics Subject Classification ID: 26A15
Mathematics Subject Classification ID: 26A21
zbMATH DE Number: 3845876
zbMATH Keywords: countable decomposition into continuous functions
zbMATH Keywords: Baire property
reviewed by: Pavel Kostyrko
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3845876