## 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.## References

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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