Real function such that restriction to any uncountable set is surjective This is a problem posed by my professor, which I don't know how to prove (or find an example) > Does there exist a function $f:\mathbb{R} \to \mathbb{R}$ such that the restriction of $f$ to any $A \subset \mathbb{R}$ with $|A| = \mathfrak{c}$ is surjective? As I've said, I haven't made significant progress. An observation is that such a function is everywhere discontinuous but Darboux. I considered the Conway Base $13$ as a possible candidate, but I'm not sure if it takes on all real values when restricted to (for example) the Cantor set.

Assume such function $f$ exists.

Let $A=\mathbb R \setminus f^{-1}(0)$. From $ f(A) = \mathbb R \setminus \\{0\\}$ follows that $A$ is not uncountable, but is mapped subjectively to a uncountable set. A contradiction.