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

$$
\forall x, y \in \mathbb{R}, \quad f(x+y)=2 f(x)+f(y)
$$
By setting $y=0$, we obtain for all $x \in \mathbb{R}, f(x)=2 f(x)+f(0)$, hence $f(x)=-f(0)$ is constant. Conversely, the only constant function that is a solution is the function identically zero.

Alternatively, swapping $x$ and $y$ allows us to notice that $f$ is constant.