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$f(x+y) = f(x) + f(y) $

I proved if $f$ is continuous at $0$ then it is continuous on $\mathbb R$ And also I proved the $f(nx) = nf(x)$ identity for rationals and integers as well.

it seems obvious $f(x) = mx$ but how do I approach from the continuity?

Lucifer
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  • @SouravGhosh I read that post but couldn't find the particular solution – Lucifer Aug 07 '22 at 03:21
  • @Lufifer I think you haven't read carefully. A solution of cauchy functional equation is continuous if it is continuous at a single point. $\$ Every continuous solution of cauchy functional equation is of the form $f(x) =mx$ – Sourav Ghosh Aug 07 '22 at 03:48
  • @SouravGhosh THANK you. I think I proved it. I took a rational number sequence which converges to a real number and used the sequential criteria to verify the solution, $=mx$ – Lucifer Aug 07 '22 at 04:18
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    See https://math.stackexchange.com/q/93816/977780 and https://math.stackexchange.com/q/356645/977780 – Sourav Ghosh Aug 07 '22 at 04:26
  • @SouravGhosh thank you I did the same! – Lucifer Aug 07 '22 at 04:42

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