$\mu(E)>0 \Rightarrow \mu(f(E))=\mu(E)$?

Question

Consider injective continuous functions $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$.

Which of these function are such that:

$\mu(E)>0 \Rightarrow \mu(f(E))=\mu(E)$?

(Isometries are clearly ok.)

Answer 1

I've discovered that is f is differentiable f satisfies my statement if and only if $det(J(f))=1 \vee -1$.