Question 89
Let $\mathbb{R}$ be the set of real numbers. We denote by $\mathcal{F}$ the set of all functions $f : \mathbb{R} \to \mathbb{R}$ such that
$$
\mathcal{F} := \{ f : \mathbb{R} \to \mathbb{R} \mid f(x + f(y)) = f(x) + f(y), \forall x, y \in \mathbb{R} \}.
$$
for every $x, y \in \mathbb{R}$. Find all rational numbers $q$ such that for every function $f \in \mathcal{F}$, there exists some $z \in \mathbb{R}$ satisfying $f(z) = qz$.
$$
\mathcal{F} := \{ f : \mathbb{R} \to \mathbb{R} \mid f(x + f(y)) = f(x) + f(y), \forall x, y \in \mathbb{R} \}.
$$
for every $x, y \in \mathbb{R}$. Find all rational numbers $q$ such that for every function $f \in \mathcal{F}$, there exists some $z \in \mathbb{R}$ satisfying $f(z) = qz$.