Page 154 of "The Axiom of Choice" by Herrlich states the proof as follows:
Assume $\mathbb{R}$ can be expressed as a direct sum $A\oplus B$ of two non-zero subgroups. The the map $f : \mathbb{R} \rightarrow \mathbb{R}$, defined by $f(a+b)=a$ for $a \in A$ and $b \in B$, is a non-continious solution of the equation $f(x+y)=f(x)+f(y)$, contradicting prop. 7.18.
(where 7.18 states Under the AD all solutions of the Cauchy-equation are continious.)
I'm a bit confused by this proof. Prehaps I don't fully understand what a non-trivial direct summand is, but mostly I am confused as to why $f(a+b)=a$ is a non-continuous solution to the Cauchy equation. Continuity of the CE implies $f(x)=f(1) \cdot x$ for all $x \in \mathbb{R}$, is this detail relevant to my understanding here?
Thanks in advance, this is my first question!