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Is $$f(x + y) = f(x) + f(y), g(x + y) = g(x) + g(y) \Leftrightarrow \ g\;o\;f = f\;o\;g $$ true?

I have a feeling that it must be true, at least the forward way, but I couldn't come up with a proof..

Any ideas?

Taylor Rendon
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Amr Ayman
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    What about $f(x)=g(x)=x^2$ as a counterexample to the reverse implication? Or any $f(x)=g(x)$? – ndhanson3 Dec 10 '20 at 00:32
  • Or $g(x) = f^{-1}(x)$ ... or if $f,g$ is an example then so is $f_1 = h\circ f\circ h^{-1}$, $g_1 = h\circ g\circ h^{-1}$ for any bijective $h$. – Greg Martin Dec 10 '20 at 00:36
  • yeah i didn't consider the reverse implication very much.. – Amr Ayman Dec 10 '20 at 00:38
  • Also, $f(x+y) = f(x)+f(y)$ doesn't imply $f:\mathbb{R} \rightarrow \mathbb{R}$ is linear (without other assumptions). –  Dec 10 '20 at 00:40
  • "Linear" has multiple definitions; $f(x+y)=f(x)+f(y)$ implies that $f$ is a linear transformation on $\Bbb R$ as a $\Bbb Q$-vector space, but not the more usual definitions of linear for such functions. – Greg Martin Dec 10 '20 at 00:43

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