Do there exist functions satisfying $f(x+y)=f(x)+f(y)$ that aren't linear?

Question

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.

Answer 1

Yes continuity is enough: You can quickly show that $f(x)=x\cdot f(1)$ for $x\in\mathbb N$, then for $x\in\mathbb Z$ and then for $x\in\mathbb Q$; assuming continuity, this implies validity for all $x\in\mathbb R$.

Any other functions only exist per Axiom of Choice: View $\mathbb R$ as a vector space over $\mathbb Q$ and take any $\mathbb Q$-linear map (which need not be $\mathbb R$-linear).

Answer 2

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 $\Bbb R^2$.