Additive function $f:\mathbb R\to \mathbb R$ that is not continuous.

Question

Let $f:\mathbb R\to \mathbb R$ an additive function, i.e. $f(x+y)=f(x)+f(y)$ for all $x,y$. In particular $f(0)=0$. I had as an exercise that if $f$ is continuous at one point, then $f(x)=xf(1)$, i.e. it's linear and continuous everywhere. Now, are they additive function that are nowhere continuous ? If yes, could someone provide an example ? I tried by contradiction : i.e. there is $x_n\to 0$ s.t. $f(x_n)\to \pm\infty $. Suppose WLOG $f(x_n)\to +\infty $. Then, for all $m$, there is $n_m$ s.t. $f(x_{n_m})>m$. Now, $$f(x_{n_m})+f(x_{n_{m+1}})=f(x_{n_m})+f(x_{n_{m+1}})>2m+1,$$ but I don't get any contradiction.