This question already has an answer here:
I created this question, but, I do not know the answer:
Is there a function $f: \mathbb R \to \mathbb R$ such that for every interval $I \subseteq \mathbb R$ we have $f(I)=\mathbb R$?
It seems to me that existence of such a function would violate almost everything that I know about analysis, but, on the other hand, it would be awesome if there is at least one such $f$, because it is known that there are many "pathological" examples in analysis. Also, many here are skilled in analysis more than I am, so I can expect an answer, curious about what it will turn to be.