Does there exist a function $f: \mathbb{R} \to \mathbb{R}$ such that for all non-empty, open, and uncountable subset $E \subseteq \mathbb{R}$, $f(E) = \mathbb{R}$?
I have been thinking about it for a bit, but really can't think of any way to prove or disprove it. I have a hunch that if a set $E \subseteq \mathbb{R}$ is open and uncountable, then there should be a bijection to $\mathbb{R}$. This may not be true, however. I'm not well versed in the more advanced concepts of cardinality, so I'm unsure if there is an uncountable set $E \subsetneq \mathbb{R}$ such that there is no bijection from $E$ to $\mathbb R$.
I'm having great difficulty imagining a function with this property. I understand that proving existence doesn't imply that construction of such a function is feasible, but I want to understand if such a function even exists.
Any help would be greatly appreciated.
