Suppose $f:\mathbb R \to \mathbb R$ satisfies the equation $f(x+y)=f(x)+f(y)$ for all $x,y\in\mathbb R$. Such a function may not be continuous, but is there still a way to extend it to a mapping from $\beta\mathbb R$ to $\beta\mathbb R$?
I am asking because I don't know much about how to extend non-continuous functions, but this function seems nice enough that maybe something will work.