What are the continuous functions $ x f(y)+y f(x)=(x+y) f(x) f(y) ? $ question - 
What are the continuous functions on $\mathbb{R}$ which are solutions of the equation
$$
x f(y)+y f(x)=(x+y) f(x) f(y) ?
$$
my try - 
by putting $y=x$ i get $f(x)=0$ or $1$ for all $x$ not equal to $0$...
now my answer is same as mention in book but i think it is incorrect because they are not valid for all $x$ ??? 
can someone tell how to fix this hole with the help of continuity ...
i know this is simple question but i want to clear my doubt...
thankyou

Suppose $f(x) =0$ for all $x \neq 0.$ Since $f$ is continuous, so $f(0)=\lim_{x \to 0} f(x)=0$. Hence $f(x)=0$ for all $x \in \mathbb{R}.$
Similarly, the other case follows. 
