Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-24T12:15:48.523Z Has data issue: false hasContentIssue false
  • English
  • Français

Almost Continuous Functions with Closed Graphs

Published online by Cambridge University Press:  20 November 2018

Andrew J. Berner
Andrew J. Berner*
Affiliation:
Department of Mathematics, University of Dallas, IrvingTexas 75061, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A function f:X → Y is almost continuous if for every x ∊ X and for each open set V ⊂ Y containing f(x), Cl(f-l(V)) is a neighborhood of x. Various conditions are given that guarantee that an almost continuous function is continuous. The main theorem states that if f:X → Y is almost continuous with a closed graph (closed in X × Y) and X and Y are complete metric spaces, then f is continuous.

Keywords

54C0854E50almost continuous functioncomplete metric space
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 25 , Issue 4 , 01 December 1982 , pp. 428 - 434
Copyright
Copyright © Canadian Mathematical Society 1982

References

1. Husain, T., Almost continuous mappings, Prace Mathematyczne 10 (1966), 1-7.Google Scholar
2. Lin, S.-Y. T. and Lin, Y.-F., On almost continuous mappings and Baire spaces, Canad. Math. Bull. 21 (1978), 183-186.Google Scholar
3. Long, P. E. and McGehee, E. E. Jr., Properties of almost continuous functions, Proc. Amer. Math. Soc. 24 (1970) 175-180.Google Scholar
4. Rose, David Alon, On Levine's decomposition of continuity, Canad. Math. Bull. 21 (1978), 477-481.Google Scholar