On sums of Darboux and nowhere constant continuous functions
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- by Krzysztof Ciesielski and Janusz Pawlikowski PDF
- Proc. Amer. Math. Soc. 130 (2002), 2007-2013 Request permission
Abstract:
We show that the property
[(P)] for every Darboux function $g\colon {\mathbb R}\to \mathbb {R}$ there exists a continuous nowhere constant function $f\colon {\mathbb R}\to \mathbb {R}$ such that $f+g$ is Darboux
follows from the following two propositions:
[(A)] for every subset $S$ of $\mathbb {R}$ of cardinality $\mathfrak {c}$ there exists a uniformly continuous function $f\colon \mathbb {R}\to [0,1]$ such that $f[S]=[0,1]$,
[(B)] for an arbitrary function $h\colon \mathbb {R}\to \mathbb {R}$ whose image $h[\mathbb {R}]$ contains a non-trivial interval there exists an $A\subset \mathbb {R}$ of cardinality $\mathfrak {c}$ such that the restriction $h\restriction A$ of $h$ to $A$ is uniformly continuous,
which hold in the iterated perfect set model.
References
- Marek Balcerzak, Krzysztof Ciesielski, and Tomasz Natkaniec, Sierpiński-Zygmund functions that are Darboux, almost continuous, or have a perfect road, Arch. Math. Logic 37 (1997), no. 1, 29–35. MR 1485861, DOI 10.1007/s001530050080
- K. Ciesielski, Set-theoretic real analysis, J. Appl. Anal. 3 (1997), no. 2, 143–190. MR 1619547, DOI 10.1515/JAA.1997.143
- Krzysztof Ciesielski, Set theory for the working mathematician, London Mathematical Society Student Texts, vol. 39, Cambridge University Press, Cambridge, 1997. MR 1475462, DOI 10.1017/CBO9781139173131
- K. Ciesielski, J. Pawlikowski, Covering property axiom CPA, version of March 2001, work in progress. (Preprint available in electronic form from Set Theoretic Analysis Web Page: http://www.math.wvu.edu/˜kcies/STA/STA.html.)
- P. Erdős, On two problems of S. Marcus, concerning functions with the Darboux property, Rev. Roumaine Math. Pures Appl. 9 (1964), 803–804. MR 181717
- Bernd Kirchheim and Tomasz Natkaniec, On universally bad Darboux functions, Real Anal. Exchange 16 (1990/91), no. 2, 481–486. MR 1112041
- Péter Komjáth, A note on Darboux functions, Real Anal. Exchange 18 (1992/93), no. 1, 249–252. MR 1205519
- A. Lindenbaum, Sur quelques propriétés des fonctions de variable réelle, Ann. Soc. Math. Polon. 6 (1927), 129–130.
- Solomon Marcus, Sur un théorème énoncé par A. Lindenbaum et démontré par W. Sierpinski, Com. Acad. R. P. Romîne 10 (1960), 551–554 (Romanian, with French and Russian summaries). MR 130328
- Arnold W. Miller, Mapping a set of reals onto the reals, J. Symbolic Logic 48 (1983), no. 3, 575–584. MR 716618, DOI 10.2307/2273449
- Juris Steprāns, Sums of Darboux and continuous functions, Fund. Math. 146 (1995), no. 2, 107–120. MR 1314977, DOI 10.4064/fm-146-2-107-120
Additional Information
- Krzysztof Ciesielski
- Affiliation: Department of Mathematics, West Virginia University, Morgantown, West Virginia 26506-6310
- Email: K_Cies@math.wvu.edu
- Janusz Pawlikowski
- Affiliation: Department of Mathematics, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland – and – Department of Mathematics, West Virginia University, Morgantown, West Virginia 26506-6310
- Email: pawlikow@math.uni.wroc.pl
- Received by editor(s): November 13, 2000
- Received by editor(s) in revised form: January 24, 2001
- Published electronically: December 27, 2001
- Additional Notes: The work of the second author was partially supported by KBN Grant 2 P03A 031 14.
- Communicated by: Alan Dow
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 2007-2013
- MSC (1991): Primary 26A15; Secondary 03E35
- DOI: https://doi.org/10.1090/S0002-9939-01-06254-2
- MathSciNet review: 1896035