  In this paper we generalize the concept of a quasi-Cauchy sequence to a
concept of a $p$-quasi-Cauchy sequence for any fixed positive integer $p$. For
$p=1$ we obtain some earlier existing results as a special case. We obtain some
interesting theorems related to $p$-quasi-Cauchy continuity, $G$-sequential
continuity, slowly oscillating continuity, and uniform continuity. It turns out
that a function $f$ defined on an interval is uniformly continuous if and only
if there exists a positive integer $p$ such that $f$ preserves $p$-quasi-Cauchy
sequences where a sequence $(x_{n})$ is called $p$-quasi-Cauchy if
$(x_{n+p}-x_{n})_{n=1}^{\infty}$ is a null sequence.
