Question 6
Let $\mathbb{R}$ be the set of real numbers. We denote by $\mathcal{F}$ the set of all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$
f(x+f(y))=f(x)+f(y)
$$
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 $f(z)=q z$.
$$
f(x+f(y))=f(x)+f(y)
$$
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 $f(z)=q z$.