$\textbf{Question:}$
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that, $f(xy)(f(x)-f(y))=(x-y)f(x)f(y)$ holds for all real number $x,y$
$\textbf{My progress:}$ If we substitute $y=1$ we get $f(x)(f(x)-xf(1))=0$ which implies
$f(x)=0$ or $f(x)=kx$ for some real $k$.
But what it doesn't say is that,whether for all $x$ it is $0$ or for all $x$ it is $kx$ aka maybe for some $x$ its $0$ and for some other its $kx.$
Then,I thought I can arrive at some sort of contradiction and prove the answer will be $kx$ for all $x$ and some real $k$.But I failed in this attempt.
In addition to helping me with this problem I would really appreciate if someone gives me some reference from where I can learn in general how to solve such tricky situations.(as I have seen such situation in some other problems too)