TITLE: what are all the continuous functions $f(x)$ that is has a domain $x>1$ and $f(ab)=f(a^2)+f(b^2)$
QUESTION [0 upvotes]: what are all the continuous functions $f(x)$ that is has a domain $x>1$ and $f(ab)=f(a^2)+f(b^2)$
the reason $f(1)$ is undefined is that $f(1)= 2\times f(1)$ and I don't want $f(1)=0$

REPLY [6 votes]: Put $a=b$ to get $f(a^{2})=2f(a^{2})$. Hence, $f(a^{2})=0$ for all $a >1$ which implies $f(x)=0$ for all $x >1$.
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