A continuous tale on continuous and separately continuous functions
type: Item
instance of: scholarly article
title: A continuous tale on continuous and separately continuous functions
zbMATH Open document ID: 1388.26010
author: Peng Zhang
published in: Real Analysis Exchange
publication date: 2018-03-12T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rae/1490752816
review text: The main objective of this survey research paper is to discuss the history and actual state of continuous and separately continuous functions problem from a real analysis perspective. The exposition includes as well some original results; the basic ones are Theorem 1. For every \(n\geq 2\), a function \(f:\mathbb{R}^n\to \mathbb{R}\) is continuous, provided it is either \((C_n^1)^*\)-continuous or \((C_n^1)\)-continuous. Theorem 2. For every \(n\geq 2\), there exists a Baire class 1 function \(h:\mathbb{R}\to \mathbb{R}^{n-1}\) such that \(T(h)\)-continuity of \(f:\mathbb{R}^n\to \mathbb{R}\) implies its continuity. Finally, simplified versions are given for old results in the area, and several open problems are formulated.
reviewed by: Mihai Turinici
Mathematics Subject Classification ID: 26B05
Mathematics Subject Classification ID: 54C08
zbMATH DE Number: 6848915
zbMATH Keywords: separate and joint continuity
zbMATH Keywords: linearly and hyperplane continuous functions
MaRDI profile type: Publication
OpenAlex ID: W2624326563
DOI: 10.14321/REALANALEXCH.41.1.0019
description: scientific article; zbMATH DE number 6848915