Sequential definitions of continuity for real functions.
type: Item
instance of: scholarly article
title: Sequential definitions of continuity for real functions.
zbMATH Open document ID: 1040.26001
author: Jeff Connor
author: Karl-Goswin Grosse-Erdmann
published in: Rocky Mountain Journal of Mathematics
publication date: 2003-12-03T00:00:00Z
full work available at URL: CONT33-1.html
review text: A function \(f:\mathbb{R}\to \mathbb{R}\) is continuous at a point \(p\in\mathbb{R}\) if and only if \(\lim_{n\to\infty} x_n= p\) implies that \(\lim_{n\to\infty}f(x_n)= f(p)\). If we replace here the usual convergence by another type of convergence we get ``a new type of continuity''. In 1948, \textit{H. Robbins} and \textit{E. C. Buck} replaced the usual convergence by the Cesàro limitation and obtained so the ``Cesàro continuity'' which gives the linearity of \(f\) [Am. Math. Mon. 53, 470--471 (1946); 55, 36 (1948), Problem 4216]. In the following years several authors continued in this study and obtained new types of continuity that led in most cases to the linearity or usual continuity of \(f\). The authors of this paper unify these types of continuity into the concept of \(G\)-continuity, where \(G\) is a linear functional. For \(G\) they take \(A\)-limitation (\(A\) being a matrix), almost convergence and statistical convergence. The relations of \(G\)-continuities to linearity and usual continuity are investigated in this paper.
Mathematics Subject Classification ID: 26A15
zbMATH DE Number: 2012050
zbMATH Keywords: Cesàro continuity
zbMATH Keywords: linearity
zbMATH Keywords: \(G\)-continuity
zbMATH Keywords: linear functional
zbMATH Keywords: almost convergence
zbMATH Keywords: statistical convergence
reviewed by: Tibor Šalát
MaRDI profile type: Publication
OpenAlex ID: W2004688262
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DOI: 10.1216/RMJM/1181069988
description: scientific article; zbMATH DE number 2012050