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On sums of Darboux and nowhere constant continuous functions


Authors: Krzysztof Ciesielski and Janusz Pawlikowski
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
Published electronically: December 27, 2001
MathSciNet review: 1896035
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Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

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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

DOI: https://doi.org/10.1090/S0002-9939-01-06254-2
Keywords: Darboux, nowhere constant, images of continuous functions
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
Article copyright: © Copyright 2001 American Mathematical Society

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