Find all continuous function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that,
\[ f(xf(y) + yf(x) )= f(f(xy)) \]
Functional Equation [Own]
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Re: Functional Equation [Own]
At last solved 
Complete solutions:
$ f(x) =\begin{cases}c &\mbox{if }x\geq a\\ a+h(x) &\mbox{if }a\geq x\geq\frac{r}{2a}\\ \frac{r}{2x}+h(x) &\mbox{if }\frac{r}{2a}\geq x\end{cases} $
Where $ 2c^{2}\geq 2ac\geq r\geq a $ and $ h:(0,a]\longrightarrow [0,\infty) $
is any continuous function with $h(a)=c-a$.
Complete solutions:
$ f(x) =\begin{cases}c &\mbox{if }x\geq a\\ a+h(x) &\mbox{if }a\geq x\geq\frac{r}{2a}\\ \frac{r}{2x}+h(x) &\mbox{if }\frac{r}{2a}\geq x\end{cases} $
Where $ 2c^{2}\geq 2ac\geq r\geq a $ and $ h:(0,a]\longrightarrow [0,\infty) $
is any continuous function with $h(a)=c-a$.
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Re: Functional Equation [Own]
Complete Solution,
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