Školjka

For every positive integer $n$ with prime factorization $n = \prod_{i = 1}^{k} p_i^{\alpha_i}$, define \[\mho(n) = \sum_{i: \; p_i > 10^{100}} \alpha_i.\]That is, $\mho(n)$ is the number of prime factors of $n$ greater than $10^{100}$, counted with multiplicity. Find all strictly increasing functions $f: \mathbb{Z} \to \mathbb{Z}$ such that \[\mho(f(a) - f(b)) \le \mho(a - b) \] for all integers $a$ and $b$ with $a > b$. \begin{flushright}\emph{(Brazil)}\end{flushright}