How to deduce the sign changes?

Question

Let $f$ be an arithmetical function. Suppose that there exists an integer set $A$ compromises the integers $n$ for which $f(n)>0$ and a set $B$ of integers $n$ such that $f(n)<0$ and a set $C$ compromises integers $n$ such that $f(n)=0$. Is there a result from number theory or an elementary result that allows us to determine the first sign changes of the sequence $(f(n))$ and to compute its number of sign changes in terms of element of the sets $A$ or $B$? Thanks in advance.