Question 74
Let $2 \mathbb{Z} + 1$ denote the set of odd integers. Find all functions $f: \mathbb{Z} \rightarrow 2 \mathbb{Z} + 1$ satisfying
$$
f(x + f(x) + y) + f(x - f(x) - y) = f(x + y) + f(x - y) \tag{1}
$$
for every $x, y \in \mathbb{Z}$.
$$
f(x + f(x) + y) + f(x - f(x) - y) = f(x + y) + f(x - y) \tag{1}
$$
for every $x, y \in \mathbb{Z}$.