  An ideal $I$ is a family of subsets of positive integers $\textbf{N}$ which
is closed under taking finite unions and subsets of its elements. A sequence
$(x_n)$ of real numbers is said to be $I$-convergent to a real number $L$, if
for each \;$ \varepsilon> 0$ the set $\{n:|x_{n}-L|\geq \varepsilon\}$ belongs
to $I$. We introduce $I$-ward compactness of a subset of $\textbf{R}$, the set
of real numbers, and $I$-ward continuity of a real function in the senses that
a subset $E$ of $\textbf{R}$ is $I$-ward compact if any sequence $(x_{n})$ of
points in $E$ has an $I$-quasi-Cauchy subsequence, and a real function is
$I$-ward continuous if it preserves $I$-quasi-Cauchy sequences where a sequence
$(x_{n})$ is called to be $I$-quasi-Cauchy when $(\Delta x_{n})$ is
$I$-convergent to 0. We obtain results related to $I$-ward continuity, $I$-ward
compactness, ward continuity, ward compactness, ordinary compactness, ordinary
continuity, $\delta$-ward continuity, and slowly oscillating continuity.
