I wonder whether there is a function $f\colon\Bbb R\to\Bbb R$ with the folowing characteristic? for every two real numbers $\alpha,\beta,\alpha\lt\beta$, $$\{f(x):x\in(\alpha,\beta)\}=\Bbb R$$ I can't say such a function does not exist, neither can I construct a example Thanks a lot!

How to find a function from reals to reals such that the image of every open interval is the whole of R? Is there one which maps rationals to rationals?

Can we have a map from $(0,1)$ to $(0,1)$ such that the image of every open interval in $(0,1)$ is all of $(0,1)$ ?

Does there exist a function $f: \mathbb{R} \rightarrow \mathbb{R} $ such that $f(a, b) = \mathbb{R} $ for every open interval $(a, b) \subset \mathbb{R} $?