Functional equation $x\space f(x^2) = f(x)$

Question

How can I logically lead to the answer from the following conditions?

$$ \left\{ \begin{align} & x \, f(x^2) = f(x) \text{ for all } x > 0, \\ & f(x) \text{ is continuous}, \\ &f(1) = 1. \end{align} \right. $$

One may easily say that $f(x)=\frac{1}{x}$ but how can we lead to that conclusion logically and exclude all other forms?

So my question is "what's the procedure to solve this equation, and what are all possible solutions?"

Thanks.

Answer 1

Consider $g(x)=xf(x)$. We have $g(x^2)=x^2f(x^2)=xf(x)=g(x)$.

By induction we have $g(x)=g(\sqrt[2^n]{x})$ for any $x > 0$. By the well known $\sqrt[2^n]{x} \to 1$, we obtain $g(x)=g(1)$ by continuity. Hence $g$ is constant, $f$ is easy to determine now.

Answer 2

Hint: Let $g(x) = xf(x)$. Then $g(x) = g(x^2)$ for all $x>0$. Use the continuity of $g$ to show that $g$ is constant.