real analysis - How to interpret the functional identity $f(a+b)=f(a)+f(b)$?

I know that $y=kx$ is definitely a solution. Other solutions may be constructed by treating the real numbers as a vector field over the rational numbers, which are pathological.

Question: Is there such an $f$ having its graph dense in $\mathbb{R}^2$?

Answer

All non-linear (or "pathological") solutions have graphs that are dense in the plane. See, eg, this SEM overview article. Or this one.