calculus - Conditions about continuous functions

Say we have $f(x+y) = f(x) + f(y) \quad \forall x,y \in \mathbb R$ and $f$ is continuous at one point at least. I wish to show there must be some $c$ such that $f(x)=cx$ for all $x$. Think I can do so by first showing $f$ is continuous everywhere I'm not sure how then let $f(q) = 1$ somehow and show that $f(q) = cq$ where $q$ is rational. But then the aim is to show for all real $x$ so I am not sure~