TITLE: Do there exist functions satisfying $f(x+y)=f(x)+f(y)$ that aren't linear?
QUESTION [14 upvotes]: Do there exist functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $f(x+y)=f(x)+f(y),$ but which aren't linear? I bet you they exist, but I can't think of any examples.
Furthermore, what hypotheses do we need to put on $f$ before no such functions exist? I feel continuity should be enough.

REPLY [2 votes]: Also, it is well known that graph $\{(x, f(x):x\in \Bbb R\}$ of every solution $f$ except $f(x)=ax$ is dense in the plane $\Bbb R^2$.