Ideal-quasi-Cauchy sequences

An ideal $I$ is a family of subsets of positive integers $\mathbf{\text{N}}$ 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 ;$\epsilon >0$ the set $\{n:\mathrm{\mid }{x}_n-L\mathrm{\mid }\ge \epsilon \}$ belongs to $I$. We introduce $I$-ward compactness of a subset of $\mathbf{\text{R}}$, the set of real numbers, and $I$-ward continuity of a real function in the senses that a subset $E$ of $\mathbf{\text{R}}$ 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 $(\mathrm{\Delta }{x}_n)$ 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.