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I am looking for the answer of this and this question.

Does there exist $f:\Bbb{R}\to \Bbb{R}$ additive onto function such that $f(F) \subset \Bbb{R}$ has the property of Baire for every $F$?

Approach 0

$$\color\white{Please strive not to post more (dupe) answers to dupes of FAQs, cf. recent site policy announcement [here](https://math.meta.stackexchange.com/questions/33508/enforcement-of-quality-standards).Instead of answering it, it would be better to look for a good duplicate target, or help the user by posting comments.You can use [this](https://approach0.xyz/search/).}$$

$$\color{red}{\boxed{\text{Darboux Property}}}\color{green}{\boxed{\text{Baire Property }}}\color{blue}{\boxed{\text{Bernstein Set}}}\color{black}{\boxed{\text{Cantor Set}}}$$ $$\color{brown}{\boxed{\text{Borel sets}}}$$ $$\color{purple}{\boxed{\text{Baire Category Theorem}}}\color{red}{\boxed{\text{Cantor Intersection Theorem}}}$$

$$\color{red}{\boxed{\LARGE\text{{Darboux Function}}}}$$

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