Question 41
Suppose that $2 \le n$ and $a_1, \dots, a_n$ are natural numbers such that $\text{gcd}(a_1, \dots, a_n) = 1$. Find all strictly increasing functions $f: \mathbb{Z} \to \mathbb{R}$ with the following property:
$$
\forall x_1, \dots, x_n \in \mathbb{Z} : f\left(\sum_{i=1}^{n} x_i a_i\right) = \sum_{i=1}^{n} f(x_i a_i).
$$
$$
\forall x_1, \dots, x_n \in \mathbb{Z} : f\left(\sum_{i=1}^{n} x_i a_i\right) = \sum_{i=1}^{n} f(x_i a_i).
$$