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

DOI:
https://doi.org/10.1090/S0002-9939-99-04955-2

Published electronically:
May 13, 1999

MathSciNet review:
1618725

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we will investigate the smallest cardinal number such that for any symmetrically continuous function there is a partition of such that every restriction is continuous. The similar numbers for the classes of Sierpinski-Zygmund functions and all functions from to are also investigated and it is proved that all these numbers are equal. We also show that and that it is consistent with ZFC that each of these inequalities is strict.

**1.**S. Baldwin,*Martin's axiom implies a stronger version of Blumberg's theorem*, Real Anal. Exchange**16**(1990-91), 67-73. MR**92b:26005****2.**A. M. Bruckner,*Differentiation of Real Functions*, CMR Series vol. 5, Amer. Math. Soc., 1994. MR**94m:26001****3.**M. Chlebík,*There are symmetrically continuous functions*, Proc. Amer. Math. Soc.**113**(1991), 683-688. MR**92b:26006****4.**J. Cicho\'{n}, M. Morayne, J. Pawlikowski, S. Solecki,*Decomposing Baire functions*, J. Symbolic Logic**56**(1991), 1273-1283. MR**92j:04001****5.**K. Ciesielski,*Set Theory for the Working Mathematician*, London Math. Soc. Student Texts**39**, Cambridge Univ. Press 1997. CMP**98:02****6.**K. Ciesielski,*Set Theoretic Real Analysis*, J. Appl. Anal.**3(2)**(1997), 143-190. (Preprint available.)**7.**K. Ciesielski, M. Szyszkowski,*A symmetrically continuous function which is not countably continuous*, Real Anal. Exchange**22**(1996-97), 428-432. (Preprint available.)

MR**97h:26002****8.**M. Kuczma,*An Introduction to the Theory of Functional Equations and Inequalities*, Polish Scientific Publishers PWN, Warsaw 1985. MR**86i:39008****9.**I. Rec{\l}aw,*Restrictions to continuous functions and Boolean algebras*, Proc. Amer. Math. Soc.**118**(1993), 791-796.) MR**93i:26003****10.**S. Shelah,*Possibly every real function is continuous on a non-meagre set*, Publications de L'Institute Mathematique - Beograd, Nouvelle Serie**57**(71) (1995), 47-60.**11.**S. Shelah, J. Stepr\={a}ns,*Decomposing Baire class 1 functions into continuous functions*, Fund. Math.**145**(1994), 171-180. MR**97c:03122****12.**J. Stepr\={a}ns,*A very discontinuous Borel function*, J. Symbolic Logic**58**(1993), 1268-1283. MR**95c:03120****13.**J. Stepr\={a}ns,*Decomposing with smooth sets*, Trans. Amer. Math. Soc., to appear. (Preprint available.) CMP**98:02**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
26A15,
03E35

Retrieve articles in all journals with MSC (1991): 26A15, 03E35

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-99-04955-2

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