Eberly College of Arts and Sciences
This note shows that if a subset S of R is such that some continuous function f from R to R has the property "f[S] contains a perfect set," then some infinitely many times differentiable function g (from R to 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 R of cardinality continuum there exists an infinitely many times differentiable function g from R to R such that g[S] contains a perfect set.
Digital Commons Citation
Ciesielski, Krzysztof, "Continuous and Smooth Images of Sets" (2012). Faculty & Staff Scholarship. 844.