Is there a nonlinear additive function on reals that is monotonic?

Question

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 $f(x)\leq f(y)$?

Answer 1

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.