An additive function is a function $f$ from $\mathbb{R}$ to $\mathbb{R}$ such that $f(x+y)=f(x)+f(y)$, for all real $x$ and $y$. Aside from the identically $0$ function, is every such function injective/surjective? Certainly, every additive function of the form $cx$ for some nonzero constant $c$ is a bijective function. So, an equivalent way of asking my question is, is every "pathological" additive function injective/surjective? If not all of them, then are at least some pathological additive functions injective/surjective?
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1Overview of basic facts about Cauchy functional equation – Anne Bauval May 17 '23 at 15:32
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Some answers in comments here: Is the function from the Cauchy functional equation, $f(x+y)=f(x)+f(y)$ injective? To summarize: using a Hamel $\Bbb Q$-basis of $\Bbb R$ (hence relying on AC), it is extremely easy to construct uncountably many "wild" solutions which are idempotent hence neither injective nor surjective, but also uncountably many "wild" involutive (hence bijective) solutions. – Anne Bauval May 17 '23 at 15:44