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On sums of Darboux and nowhere constant continuous functions

Authors:	Krzysztof Ciesielski Janusz Pawlikowski

Institution:	Department of Mathematics, West Virginia University, Morgantown, West Virginia 26506-6310 ; Department of Mathematics, University of Wroclaw, pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland -- and -- Department of Mathematics, West Virginia University, Morgantown, West Virginia 26506-6310

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 is Darboux follows from the following two propositions: (A) for every subset of $\mathbb{R}$ of cardinality $\mathfrak{c}$ there exists a uniformly continuous function $f\colon\mathbb{R}\to0,1]$ such that , (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 to is uniformly continuous, which hold in the iterated perfect set model.

Keywords:	Darboux nowhere constant images of continuous functions

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