Cauchy's Problem

Question

I am looking at the Cauchy's functional equation here: http://en.wikipedia.org/wiki/Cauchy's_functional_equation.

Could someone help me on how to generalize the Cauchy's equation to $x \in \mathbb{R}$?

I know all the steps leading to the proof that the only possible functional equation $f(x)$ such that $f(x+y)=f(x)+f(y)$ for all $x \in \mathbb{Q}$ is the function $Cx$, where $x=f(1)$.

Thanks,

Answer 1

Hint: $f$ is continuous, for every irrational $s \in \mathbb Q'$ we have a sequence of rational numbers $r_n$ which converges to $s$, by continuity $f(r_n) \to f(s)$