Open Access
2011/2012 Continuous and Smooth Images of Sets
Krzysztof Chris Ciesielski, Togo Nishiura
Real Anal. Exchange 37(2): 305-314 (2011/2012).


This note shows that if a subset \(S\) of \(\mathbb{R}\) is such that some continuous function \(f\colon\mathbb{R}\to\mathbb{R}\) has the property “\(f[S]\) contains a perfect set,” then some \(\mathcal{C}\,^\infty\) function \(g\colon\mathbb{R}\to\mathbb{R}\) has the same property. Moreover, if \(f[S]\) is nowhere dense, then the \(g\) can have the stronger property “\(g[S]\) is perfect.” The last result is used to show that it is consistent with ZFC (the usual axioms of set theory) that for each subset \(S\) of \(\mathbb{R}\) of cardinality \(\mathfrak{c}\) (the cardinality of the continuum) there exists a \(\mathcal{C}\,^\infty\) function \(g\colon \mathbb{R}\to\mathbb{R}\) such that \(g[S]\) contains a perfect set.


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Krzysztof Chris Ciesielski. Togo Nishiura. "Continuous and Smooth Images of Sets." Real Anal. Exchange 37 (2) 305 - 314, 2011/2012.


Published: 2011/2012
First available in Project Euclid: 15 April 2013

zbMATH: 1283.26004
MathSciNet: MR3080593

Primary: 26A24
Secondary: 03E35 , 26A03

Keywords: images , infinitely smooth , perfect sets

Rights: Copyright © 2011 Michigan State University Press

Vol.37 • No. 2 • 2011/2012
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