This question already has an answer here:
Is there a function $f:\mathbb R\to\mathbb R$ such that $f(x+y)=f(x)+f(y)$ which is not continuous? I have proved that if it's continuous in one point $a\in\mathbb R$ then it's continuous on all $\mathbb R$, but I didn't find such a function which is not continuous everywhere. Therefore I tried to prove that all function of this form is continuous at $x=0$ but with no success. I think that if such a function exist it would be of the form $f(x)=...$ if $x\in\mathbb Q$ and $f(x)=...$ if $x\in\mathbb R\backslash\mathbb Q$ but I didn't find it.