Find all functions $f$ such that $f\left(x^2-y^2\right)=(x-y)\big(f(x)+f(y)\big)$. 
Find all functions $f:\mathbb R\to\mathbb R$ such that
$$f\left(x^2-y^2\right)=(x-y)\big(f(x)+f(y)\big)\text.$$

I have derived these clues:

*

*$f(0)=0$;

*$f(x^2)=xf(x)$;

*$f(x)=-f(-x)$.

But now I am confused. I know solution will be $f(x)=x$, but I don't know how to prove this.

$f(x)+f(x+1)=f(2x+1)=(2x+1)(f(x+1)+f(-x))$.  
That can be rearranged to 
$$\frac{f(x)}{x}=\frac{f(x+1)}{x+1}$$
so, for example, $f(n)=nf(1)$
