$f(x) = x f(\frac{1}{x})$ for $x>0$

Question

For $x>0$, $f(x)>0$ and continuous. Also $$f(x)=xf\bigg(\frac{1}{x}\bigg).$$ The function $f(x)=x \ \text{for} \ x\in(0,1]$ and $f(x)=1 \ \text{for} \ x>1$ certainly satisfies the above conditions. Is this the unique function satisfying the above conditions? Or what is another example function?

Thanks

Answer 1

\begin{align} &f(x)=xf\Big(\frac 1 x\Big), f: \mathbb{R^{+}} \to \mathbb{R^{+}}\text{ and Continuous.} \\ \ \\ &\text{if } f \equiv c: \\ &c=xc \Rightarrow c=0, \text{ Contradiction.(By $f: \mathbb{R^+} \to \mathbb{R^+}$.)} \\ \therefore \; & f \not\equiv c. \\ \end{align} This shows that the function isn't constant...

I just thought about another form of this function and found another one.

I think these 2 functions satisfy the F.E...

I thought very shortly, so nevermind this one.