TITLE: The No. of Continuous function satisfying the condition $xf(y)+yf(x)=(x+y)f(x)f(y)$
QUESTION [6 upvotes]: Could anyone tell me? 
The No. of Continuous function satisfying the condition $$xf(y)+yf(x)=(x+y)f(x)f(y)$$ is 
$1,2,3,$ or  none of them?
or give me hints please.

REPLY [10 votes]: Assuming your functions are from $\mathbb{R} \to \mathbb{R}$
$$
f(1)+f(1) = (1+1)f(1)f(1) \Rightarrow f(1) = 0 \text{ or } 1
$$
If $f(1) = 1$, then for any $x\neq 0$
$$
x + f(x) = (x+1)f(x) \Rightarrow f(x) = 1
$$
By continuity, $f\equiv 1$
Similarly, if $f(1) = 0$, then $f\equiv 0$.
Hence, the answer is 2.