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A function $f\colon\mathbb R\to\mathbb R$ is called additive if, for all $x,y\in\mathbb R$, $f(x+y)=f(x)+f(y)$. Obviously, every linear function on $\mathbb R$ is additive.
However, if we assume axiom of choice, it is not difficult to show that there exists an additive function that is not linear.
Question Assuming axiom of choice, is there a nonlinear additive function $f$ such that $x\leq y$ implies $f(x)\leq f(y)$?
No. By standard arguments, such a function would need to equal multiplication by a constant when restricted to $\mathbb Q$, and monotonicity now fully determines its value everywhere else.
