Find all the function that satisfy : $$f\left(\frac{xf(y)}{2}\right)+f\left(\frac{yf(x)}{2}\right)=4xy$$ I only find $f(0)=0$ but I can't prove $f(x)=2x$

  • 4
    $\begingroup$ $f(x)=-2x$ is another solution. $\endgroup$ Feb 25, 2013 at 14:42
  • $\begingroup$ Is $f$ a continuous function? And what is the domain, $\mathbb{R}$ or $\mathbb{C}$? $\endgroup$
    – Yimin
    Feb 25, 2013 at 15:57

1 Answer 1


If your $f$ is differentiable at $x=0$. Then

Differentiate the equation for $x$ at $x=0$


Take $x=0$. Since $f(0)=0$. Say $A = f'(0)$.


Which is

$f(y)= \dfrac{8y-A^2y}{A}$

$f'(y) = \dfrac{8-A^2}{A}$,take $y=0$, $f'(0)=A=\dfrac{8-A^2}{A}$.

$A=2$ or $A=-2$.

This is for smooth case. If not differentiable at $x=0$, then I have no idea.


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