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Proceedings of the American Mathematical Society

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



Decomposing symmetrically continuous
and Sierpinski-Zygmund functions
into continuous functions

Author: Krzysztof Ciesielski
Journal: Proc. Amer. Math. Soc. 127 (1999), 3615-3622
MSC (1991): Primary 26A15; Secondary 03E35
Published electronically: May 13, 1999
MathSciNet review: 1618725
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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 Sierpinski-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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Keywords: Decomposition number, symmetrically continuous functions, Sierpi\'nski-Zygmund functions
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
Article copyright: © Copyright 1999 American Mathematical Society