Real Analysis Exchange

Continuous and Smooth Images of Sets

Krzysztof Chris Ciesielski and Togo Nishiura

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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.

Article information

Real Anal. Exchange, Volume 37, Number 2 (2011), 305-314.

First available in Project Euclid: 15 April 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15]
Secondary: 03E35: Consistency and independence results 26A03: Foundations: limits and generalizations, elementary topology of the line

infinitely smooth images perfect sets


Ciesielski, Krzysztof Chris; Nishiura, Togo. Continuous and Smooth Images of Sets. Real Anal. Exchange 37 (2011), no. 2, 305--314.

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