2020 Types of Convergence Which Preserve Continuity
Simon Reinwand
Real Anal. Exchange 45(1): 173-204 (2020). DOI: 10.14321/realanalexch.45.1.0173


In the first part of this paper we investigate four types of convergence of sequences of functions in metric spaces which preserve continuity. Besides a consideration of locally, quasi and continuously uniform convergence, we introduce the notion of semi uniform convergence. We discuss how all types of convergence are related both to each other and to pointwise convergence, and illustrate their behavior by examples. Moreover, we show how some of the types of convergence can be used to characterize compactness of the domains the functions under consideration live in. In the second part we investigate sequences of composition operators in the space \(BV\) of functions of bounded variation in the sense of Jordan. We give criteria under which such sequences converge locally uniformly and semi uniformly and present a new and short proof for the fact that composition operators which map the space \(BV\) into itself are automatically continuous.


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Simon Reinwand. "Types of Convergence Which Preserve Continuity." Real Anal. Exchange 45 (1) 173 - 204, 2020. https://doi.org/10.14321/realanalexch.45.1.0173


Published: 2020
First available in Project Euclid: 9 May 2020

zbMATH: 07211609
Digital Object Identifier: 10.14321/realanalexch.45.1.0173

Primary: 26A15 , 26A45 , 40A30
Secondary: 54E45

Keywords: compact metric spaces , continuity of composition operators , Functions of bounded variation , locally, quasi, semi and continuously uniform convergence , pointwise convergence , sequences of composition operators , sequences of continuous functions in metric spaces

Rights: Copyright © 2020 Michigan State University Press


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Vol.45 • No. 1 • 2020
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