Find the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $\forall x, y \in \mathbb{R}, f(x+y)=x+f(y)$.
The half-symmetry allows us to write $\forall x, y \in \mathbb{R}, y+f(x)=x+f(y)$, i.e., $f(x)-x=f(y)-y$, which means that $f$ is of the form $x \mapsto x+a$, for $a \in \mathbb{R}$. Conversely, all functions of this form are suitable.