Is there a function $f:\mathbb{R}\rightarrow\mathbb{R}$ that maps every subset of cardinality $|\mathbb{R}|$ to the whole real line?

Functions like Conway's base-13 function map each open set to the whole real line. Functions like this one I think map every set of positive Lebesgue measure to the real line but I don't know how to prove it.Can this result be extended in the way described in the question? If it does not, then the only other functions with this property use Axiom of Choice.