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 $\mathrm{ZFC}$.


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