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ON SEPARATE CONTINUITY AND SEPARATE CONVEXITY: A SYNTHETIC TREATMENT FOR FUNCTIONS AND SETS

Part of: General convexity Functions of several variables Basic constructions

Published online by Cambridge University Press:  17 July 2023

METIN UYANIK Open the ORCID record for METIN UYANIK [Opens in a new window]  and
M. ALI KHAN Open the ORCID record for M. ALI KHAN [Opens in a new window]
METIN UYANIK*
Affiliation:
School of Economics, University of Queensland, Brisbane, Queensland 4072, Australia
M. ALI KHAN
Affiliation:
Department of Economics, Johns Hopkins University, Baltimore, MD 21218, USA e-mail: akhan@jhu.edu
*
e-mail: m.uyanik@uq.edu.au
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Abstract

This paper relies on nested postulates of separate, linear and arc-continuity of functions to define analogous properties for sets that are weaker than the requirement that the set be open or closed. This allows three novel characterisations of open or closed sets under convexity or separate convexity postulates: the first pertains to separately convex sets, the second to convex sets and the third to arbitrary subsets of a finite-dimensional Euclidean space. By relying on these constructions, we also obtain new results on the relationship between separate and joint continuity of separately quasiconcave, or separately quasiconvex functions. We present examples to show that the sufficient conditions we offer cannot be dispensed with.

Keywords

convexityseparate convexitycontinuityseparate continuitylinear continuityarc-continuityrestricted continuity

MSC classification

Primary: 26B05: Continuity and differentiation questions
Secondary: 52A99: None of the above, but in this section 54B99: None of the above, but in this section
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 109 , Issue 3 , June 2024 , pp. 541 - 551
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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