  A function $f$ from an Abelian group $(A,+)$ to an Abelian group $(B,+)$ is
$(n, m, S)$ zero-difference (ZD), if $S=\{\lambda_\alpha \mid \alpha \in
A\setminus\{0\}\}$ where $n=|A|$, $m=|f(A)|$ and $\lambda_\alpha=|\{x \in A
\mid f(x+\alpha)=f(x)\}|$. A function is called zero-difference balanced (ZDB)
if $S=\{\lambda\}$ where $\lambda$ is a constant number. ZDB functions have
many good applications. However it is point out that many known zero-difference
balanced functions are already given in the language of partitioned difference
family (PDF). The problem that whether zero-difference "not balanced" functions
still have good applications as ZDB functions, is investigated in this paper.
By using the change point technic, zero-difference functions with good
applications are constructed from known ZDB function. Then optimal difference
systems of sets (DSS) and optimal frequency-hopping sequences (FHS) are
obtained with new parameters. Furthermore the sufficient and necessary
conditions of these being optimal, are given.
