ISL 2015 N8

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 $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that $$\mho(f(a)-f(b)) \leqslant \mho(a-b) \text { for all integers } a \text { and } b \text { with } a>b .$$