Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying

$$
\forall x, y \in \mathbb{R}, \quad f(x+y)=f(x)-f(y)
$$
It suffices to set $x=y$, and we obtain

$$
f(2 x)=f(x)-f(x)=0
$$

For $z=2 x, z$ runs through all $\mathbb{R}$ and we have $f(z)=0$. The only solution is therefore the zero function, which indeed satisfies the initial equation.