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2004-2005 More about Sierpiński-Zygmund uniform limits of extendable functions
Harvey Rosen
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Real Anal. Exchange 30(1): 129-136 (2004-2005).


Let $SZ$, $D$, $Ext$, and $\overline{Ext}$ denote respectively the spaces of Sierpiński-Zygmund functions, Darboux functions, extendable connectivity functions, and uniform limits of sequences of extendable connectivity functions, with the metric of uniform convergence on them. We show that the subspaces $SZ \cap D$ and $SZ \cap \overline{Ext}$ are each porous in the space $SZ$, but $SZ \cap \overline{Ext}$ is not porous in the space $\overline{Ext}$. We also show that every real function can be expressed as a sum of two Sierpiński-Zygmund functions one of which belongs to $\overline{Ext}$. Ciesielski and Natkaniec showed in 1997 that if $\R$ is not the union of less than $\mathfrak c$-many nowhere dense subsets, then there exist Sierpiński-Zygmund bijections $f,g:\mathbb{R}\to\mathbb{R}$ such that $f^{-1} \notin SZ$ and $g^{-1}\in SZ$, but here we can additionally have $f$ and $g$ belonging to $\overline{Ext}$


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Harvey Rosen. "More about Sierpiński-Zygmund uniform limits of extendable functions." Real Anal. Exchange 30 (1) 129 - 136, 2004-2005.


Published: 2004-2005
First available in Project Euclid: 27 July 2005

zbMATH: 1066.26003
MathSciNet: MR2126800

Primary: 26A15 , 54C35‎

Keywords: Darboux function , inverse function , porosity , Sierpiński-Zygmund function , uniform limit of extendable connectivity functions

Rights: Copyright © 2004 Michigan State University Press

Vol.30 • No. 1 • 2004-2005
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