Continuous and Smooth Images of Sets

Abstract

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.