I need to solve the following functional equation: $xf\left(\frac{1}{x}\right)=f(x)$.
By observation, $f(x)=\sqrt{x}$ is a solution, but I don't know how to find a general solution, or show that this is the only solution. I've tried forming a differential equation, and just trying out different values for x, but so far have nothing.
Let $g(x)=f(x)/\sqrt{x}$ and we have $g(x)=g(1/x)$. We only need to consider $g$.
However, $g$ is free on $[1,+\infty)$, which means you can randomly set the value $g(x)$ there and just extend it onto $(0,1)$ according to the equation above.