Is there any positive function $f$ such that $f(x)f(y)\leq |x-y|$ for every $x$ rational and $y$ irrational?
Is there a positive function $f$ on real line such that $f(x)f(y)\le|x-y|, \forall x\in \mathbb Q , \forall y \in \mathbb R \setminus \mathbb Q$?