Additive functions on the real line

A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is said to be additive if $f(x + y) = f(x) + f(y)$ for every $x, y \in \mathbb{R}$. Typical examples of additive functions are the linear ones, i.e. $f(x) = ax$ where $a$ is a real constant. One question is that are they the only possibilities? If we further require that $f$ to be continuous, then the answer is positive. But in general, the answer is no and one can use basic linear algebra to construct counter-examples.

The trick is that we can regard $\mathbb{R}$ as a vector space over $\mathbb{Q}$ (and the dimension is infinite, exercise). Let $\beta$ be a $\mathbb{Q}$-basis of $\mathbb{R}$ which contains 1. This can be done by a standard application of Zorn's Lemma. Define $f: \mathbb{R} \rightarrow \mathbb{R}$ on $\beta$ as below and extend $\mathbb{Q}$-linearly. Take $f(1) = 1$ and $f(v) = 0$ whenever $1 \neq v \in \beta$. By definition, $f$ is $\mathbb{Q}$-linear, so it is additive. To see that $f$ is not $\mathbb{R}$-linear, let $0 \neq x \in \mathbb{R}$ be spanned $\mathbb{Q}$-linearly by basis vectors in $\beta$ excluding 1. Then $f(x) = 0$ by construction but $xf(1) = x \neq f(x)$.