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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.

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    $\begingroup$ What sort of properties would you want the extension to have? $\endgroup$ – Eric Wofsey Aug 31 '16 at 7:11
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    $\begingroup$ 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 ? $\endgroup$ – user171326 Aug 31 '16 at 7:11
  • $\begingroup$ What about the identity function or the $0$ function? Both satisfy $f(x + y)= f(x) + f(y)$. $\endgroup$ – Oles Wohnzimmer Aug 31 '16 at 7:16
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    $\begingroup$ @OlesWohnzimmer But I want a general way of extending. As for properties maybe $\beta f$ can satisfy the same equation? Where $p+q=\{A+B:A\in p,B\in q\}$? $\endgroup$ – Forever Mozart Aug 31 '16 at 7:21
  • $\begingroup$ @ForeverMozart Sorry..what is this notation $\beta \mathbb R$ ?Compacatification? $\endgroup$ – Dontknowanything Aug 31 '16 at 7:25

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