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 $f(1)=0$
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You can't let $b=0$ because $0$ is not in the domain. – Pedro Amaral Jan 17 '21 at 23:44
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1I think it would slightly more interesting to consider $f(ab) = \frac{f(a^2) +f(b^2)}{2}$. That way, there is a formal similarity to the inequality $ab \le \frac{a^2 + b^2}2$. – User8128 Jan 18 '21 at 01:21
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@User8128 okay I'll make a new question. – Jan 18 '21 at 15:59
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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 $x >1$.
Kavi Rama Murthy
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