The No. of Continuous function satisfying the condition $xf(y)+yf(x)=(x+y)f(x)f(y)$

Question

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.

Answer 1

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.