Question 173
Let $\mathcal{F}$ be the set of all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $f(x + f(y)) = f(x) + f(y)$ for all real numbers $x$ and $y$. Determine all real numbers $r$ satisfying the following condition: For every function $f$ in $\mathcal{F}$, there exists some real number $z$ such that $f(z) = rz$.