On sums of Darboux and nowhere constant continuous functions

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 non-trivial interval there exists an A ⊂ 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. ∗AMS classification numbers: Primary 26A15; Secondary 03E35.