6-147 Find all functions $f: R \rightarrow R$ that satisfy
$x f(y)+y f(x) \equiv(x+y) f(x) f(y), x, y \in R$.
[Solution] In the identity given in the problem, let $x=y=1$, then we get $2 f(1)=2(f(1))^{2}$, i.e., $f(1)=0$ or $f(1)=1$. We will examine these two cases separately as follows:
(1) If $f(1)=0$, then by setting $y=1$ in the identity, we get $f(x) \equiv 0$.
(2) If $f(1)=1$, then by still setting $y=1$, we obtain
$$
f(x)+x \equiv(x+1) f(x) \text {. }
$$

i.e., $x(f(x)-1) \equiv 0$.
Therefore, for any $x \neq 0$, we have $f(x)=1$. Thus, there are two possible functions for $f(x)$: either $f(x) \equiv 0$, or

conditions.