measure theory - Prove that if a particular function is measurable, then its image is a rect line

I´m really stuck with this problem of my homework. I don´t have any idea, how to begin.

Let $f$ be a function, $f:\mathbb{R}\rightarrow \mathbb{R}$, such that $f(x+y)=f(x)+f(y)$ , $\forall x,y\in\mathbb{R}$. Prove that if f is Lebesgue-measurable, then there exists a $c\in\mathbb R$, such that $f(x)=cx$.

I have the next idea, $\forall \alpha\in \mathbb R$, $f^{-1}((-\infty,\alpha))$, (where $f^{-1}$ denotes the preimage function) is measurable. And somehow I want to end with that the image of such set is $(-\infty,c\alpha)$. Any idea is welcome.