Raja Oktovin wrote: Find all function $ f: \mathbb{R} \rightarrow \mathbb{R}$ such that \[ f(x + y)(f(x) - y) = xf(x) - yf(y) \] for all $ x,y \in \mathbb{R}$. $ y = 0$ gives $ f(x)^2 = xf(x)$ then $ f(x) = x$ or $ f(x)=0$ we suppose that $ f(a)=a$ for some $ a\neq 0$ if $ x+y=a$ we have $ f(x+y)(f(x)-y)=xf(x)-yf(y)=f(x+y)(-f(y)+x)$ gives $ f(x)=x$ and $ f(y)=y$ then $ f(x)=x$ for all $ x$ if $ a$ don't exist we will have $ f(x)=0$