# Stone-Cech extension of a function satisfying $f(x+y)=f(x)+f(y)$?

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.

• What sort of properties would you want the extension to have? – Eric Wofsey Aug 31 '16 at 7:11
• What are you asking for property of such $\beta f$ ? Obviously it cannot be continous since it restriction will not necessarly be continous. So what property do you want ? – user171326 Aug 31 '16 at 7:11
• What about the identity function or the $0$ function? Both satisfy $f(x + y)= f(x) + f(y)$. – Oles Wohnzimmer Aug 31 '16 at 7:16
• @OlesWohnzimmer But I want a general way of extending. As for properties maybe $\beta f$ can satisfy the same equation? Where $p+q=\{A+B:A\in p,B\in q\}$? – Forever Mozart Aug 31 '16 at 7:21
• @ForeverMozart Sorry..what is this notation $\beta \mathbb R$ ?Compacatification? – Dontknowanything Aug 31 '16 at 7:25