Question 63
A function $f: \mathbb{R} \to \mathbb{R}$ satisfies
$$
f(f(a)) = f(a) \quad \text{and} \quad f(a+b) = f(a) + f(b)
$$
for all real numbers $a, b$. Prove that, for all real $x$, there exists a unique $y$ such that $f(y) = 0$ and $x = y + f(z)$ for some real $z$.
$$
f(f(a)) = f(a) \quad \text{and} \quad f(a+b) = f(a) + f(b)
$$
for all real numbers $a, b$. Prove that, for all real $x$, there exists a unique $y$ such that $f(y) = 0$ and $x = y + f(z)$ for some real $z$.