1

Given $f:\mathbb{R}\rightarrow\mathbb{R}$, and the function $f$ satisfies $f(x+y)=f(x)+f(y)$ for any $x,y\in S$. Can we say that this function $f$ must be continuous?

I think it is false, but couldn't give an example. Can someone tell me whether it is true or not? Thank you very much.

Jose Arnaldo Bebita Dris
  • 1
python3
  • 3,494
  • No.consider Micah's link.on the other hand assuming continuity, it must be exponential – BigM Mar 02 '15 at 22:23
  • @Brian: You're absolutely right. You should have the dupe hammer powers to reopen and reclose. And you should do that. – Asaf Karagila Mar 02 '15 at 22:24
  • @Asaf: I didn’t realize that the hammer extended to reopening; thanks! – Brian M. Scott Mar 02 '15 at 22:26
  • 1
    The answer may be a matter of opinion! If you take the AXiom of Choice to be true, the answer is that $f$ need not be continuous. If you take the Axiom of Determinacy to be true, the answer is that $f$ must be continuous. – André Nicolas Mar 02 '15 at 22:32

0 Answers0