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what are all the continuous functions $f(x)$ that is has a domain $x>0$ and $f(ab)=\frac{f(a^2)+f(b^2)}{2}$.

this question was changed because the answer to the original problem what are all the continuous functions $f(x)$ that is has a domain $x>1$ and $f(ab)=f(a^2)+f(b^2)$ was trivial and User$8128$ pointed out

I 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 $ab \le \frac{a^2 + b^2}2$

J.G.
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    $\log (x)$ satisfies this, so you might try to prove that the only non-constant solutions are logs to various bases. – lulu Jan 18 '21 at 16:11
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    Quite obviously, $f(x)=c_1+c_2\ln x$. –  Jan 18 '21 at 16:17
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    The recent edit from $C=2$ to general $C$ is pointless. If $C\ne2$ then $f(a^2)=\tfrac2Cf(a^2)$ so $f$ is identically $0$. – J.G. Jan 18 '21 at 16:21

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Write $g(x):=f(e^x)$ to restate the problem as $g((x+y)/2)=(g(x)+g(y))/2$, with $g$ continuous on $\Bbb R$. Finish with this question, which proves the suitable $g$ are precisely the linear $g$, in accord with @ProfessorVector's characterization of the suitable $f$.

J.G.
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It's $\log(x)$ it follows the same structure as logs

$\log(ab)=\log(a^2)/2+\log(b^2)/2$

$\log(a)/2=\log(a^{0.5})$

$\log(a^2)/2=\log(a)$

$\log(ab)=\log(a)+\log(b)$

definition

$\log$ is in any base

V.G
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