Define $A=\sup \limits\_{f} \{m^\*(f[0, 1])\}$, here $f$ are all the functions which satisfy that
$f: [0, 1] \rightarrow [0, 1]$; $\forall x, f(x)-x \in \mathbb{Q}; \forall x, y, x-y \in \mathbb{Q} \rightarrow f(x)=f(y)$.


What is the value of $A$? Obviously, $0<A \leq 1$.



What is the value of this supremum?
Your conditions may be restated as: $f$ induces a choice function for the partition of $[0,1]$ defined by the equivalence $x\sim y$ iff $x-y\in \mathbb Q$. So the image of $f$ is a Vitali set (a choice of representatives for that partition). Therefore $A=1$ because there are Vitali set of outer measure $1$. Here are
constructions:


<https://math.stackexchange.com/questions/14591/vitali-type-set-with-given-outer-measure>
<https://math.stackexchange.com/questions/157532/vitali-set-of-outer-measure-exactly-1>


