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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Decomposing symmetrically continuous and Sierpinski-Zygmund functions into continuous functions
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by Krzysztof Ciesielski PDF
Proc. Amer. Math. Soc. 127 (1999), 3615-3622 Request permission

Abstract:

In this paper we will investigate the smallest cardinal number $\kappa$ such that for any symmetrically continuous function $f\colon \mathbb {R}\to \mathbb {R}$ there is a partition $\{X_\xi \colon \xi <\kappa \}$ of $\mathbb {R}$ such that every restriction $f\restriction X_\xi \colon X_\xi \to \mathbb {R}$ is continuous. The similar numbers for the classes of Sierpiński-Zygmund functions and all functions from $\mathbb {R}$ to $\mathbb {R}$ are also investigated and it is proved that all these numbers are equal. We also show that $\mathrm {cf}(\mathfrak {c})\leq \kappa \leq \mathfrak {c}$ and that it is consistent with ZFC that each of these inequalities is strict.
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Additional Information
  • Received by editor(s): November 23, 1997
  • Received by editor(s) in revised form: February 18, 1998
  • Published electronically: May 13, 1999
  • Additional Notes: The author was partially supported by NATO Collaborative Research Grant CRG 950347 and 1996/97 West Virginia University Senate Research Grant.
  • Communicated by: Alan Dow
  • © Copyright 1999 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 127 (1999), 3615-3622
  • MSC (1991): Primary 26A15; Secondary 03E35
  • DOI: https://doi.org/10.1090/S0002-9939-99-04955-2
  • MathSciNet review: 1618725