how to find all functions such that $f\left( x^{2} - y^{2} \right) = ( x - y )( f(x) + f(y) )$ Find all function $f:\mathbb R\to\mathbb R$ such that $f\left( x^{2} - y^{2} \right) = ( x - y )( f(x) + f(y) )$.
My try:
If $ x=y=0$ then $f(0)=0$ and 
if $x\leftarrow\frac{x+1}{2}$ and $y\leftarrow \frac{x-1}{2}$, then 
$f(x)=f\left( \frac{x+1}{2}  \right)+f\left( \frac{x-1}{2}  \right)$.
But how to find all functions?

We can solve this one in the same manner as here.
So, it is easy to see that $f(0)=0$ and $f$ is odd.
Let $f(1)=0$. If we put $y=1$ we get $$f(x^2-1) = (x-1)f(x)$$ and if we put $-x$ instead of $x$ we get $$f(x^2-1) = -(x+1)f(-x) = (x+1)f(x)$$ So we have $$(x-1)f(x)= (x+1)f(x)\implies f(x)=0\;\;\;\forall x$$ and we are done.
