On sums of Darboux and nowhere constant continuous functions

We show that the property

(P)

for every Darboux function $g\colon{\mathbb R}\to\mathbb{R}$ there exists a continuous nowhere constant function $f\colon{\mathbb R}\to\mathbb{R}$ such that $f+g$ is Darboux

follows from the following two propositions:

(A)

for every subset $S$ of $\mathbb{R}$ of cardinality $\mathfrak{c}$ there exists a uniformly continuous function $f\colon\mathbb{R}\to[0,1]$ such that $f[S]=[0,1]$,

(B)

for an arbitrary function $h\colon\mathbb{R}\to\mathbb{R}$ whose image $h[\mathbb{R} ]$ contains a non-trivial interval there exists an $A\subset\mathbb{R}$ of cardinality $\mathfrak{c}$ such that the restriction $h\restriction A$ of $h$ to $A$is uniformly continuous,

which hold in the iterated perfect set model.