I'm looking for a function $f:\mathbb{R}\longrightarrow\mathbb{R}$ which is surjective on every open subset of the real line.
Obviously this calls for an inductive construction. But how do I do that ensuring I don't break anything later in the process? I was thinking about putting all the open subsets into a sequence of length continuum, and then setting $f_0$ surjective from $U_0$ onto $\mathbb{R}$ and $f_{\alpha}$ surjective from $U_{\alpha}\setminus\bigcup\limits_{\gamma < \alpha} U_{\gamma}$ onto $\mathbb{R}$ anyhow I want (if it is nonempty). I thought this would be a well defined construction, as this difference would always be non-countable if not empty. But now I see that there is no contradiction in saying that, say, $U_5=(0,1),\ U_6=(1,2),\ U_7=(\frac{1}{2},\frac{3}{2})$ and in that case... I'm not sure what to do.
Please, help.
