Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $x$ and $y$:

$$
f\left(x^{2}-y^{2}\right)=x f(x)-y f(y)
$$
For $y=0$, we have $f\left(x^{2}\right)=x f(x)$, which particularly gives the fact that $f$ is odd.

Furthermore, this gives $f\left(x^{2}-y^{2}\right)=f\left(x^{2}\right)+f\left(-y^{2}\right)$, thus $f(a+b)=f(a)+f(b)$ for $a \geqslant 0$ and $b \leqslant 0$. The oddness allows us to have $f(a+b)=f(a)+f(b)$ in the other cases.

We take $x=t+1$ and $y=t$, to get

$$
2 f(t)+f(1)=f(2 t+1)=f\left((t+1)^{2}-t^{2}\right)=(t+1) f(t+1)-t f(t)=f(t)+(t+1) f(1)
$$

Thus $f(t)=t f(1)$ and $f$ is a linear function, we verify that they work.