On sums of Darboux and nowhere constant continuous functions

Abstract

We show that the property (P) for every Darboux function g: R → R there exists a continuous nowhere constant function f: R → R such that f + g is Darboux follows from the following two propositions: (A) for every subset S of R of cardinality c there exists a uniformly continuous function f: R → [0, 1] such that f[S] = [0, 1], (B) for an arbitrary function h: R → R whose image h[R] contains a nontrivial interval there exists an A C R of cardinality c such that the restriction h | A of h to A is uniformly continuous, which hold in the iterated perfect set model.