# Concerning Luzin-(N)-property

Definition: a function $$f:\mathbb{R}\to \mathbb{R}$$ has Luzin-(N)-Property if $$f$$ maps any null set to a null set.

By https://www.encyclopediaofmath.org/index.php/Luzin-N-property, it is known that if $$f$$ has Luzin-(N)-property and measurable, then for almost every real $$x$$, $$f^{-1}(x)$$ is countable. Now I have the following question:

Question Is there a function $$f$$ having Luzin-(N)-property and a nonnull set $$A$$ so that for any $$x\in A$$, $$f^{-1}(x)$$ is uncountable?

The question has a positive answer under certain set theoretical assumptions. But I want an answer within $$\mathrm{ZFC}$$.