Let $f:\mathbb R\to\mathbb R$ be a function with the property that $f(a+b)=f(a)+f(b)$ for all real numbers $a$ and $b$. Assume that the limit as $x\to 0$ of $f(x)$ is equal to some real number $L$. Show $L=0$.
I started to attempt to use the epsilon-delta definition of continuity, but I'm stuck. Please help!
Edit: While I know that the functions for which this is true are, for example, $f(x)=cx$, I can't assume anything that is not given.