There is an example of a function that is unbounded on every open set. Just take $f(n/m) = m$ for coprime $n$ and $m$ and $f(irrational) = 0$.


I want to generalize this in a way to get a function which is not just unbounded on every open set, but whose range is equal to $\mathbb{R}$ on every open set. The latter construction clearly doesn't work.


I'm interested whether such function exists and if it exists is there any constructive way to define it?



Function with range equal to whole reals on every open set
See [Conway's base 13 function](http://en.wikipedia.org/wiki/Conway_base_13_function).


