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    Ideal-quasi-Cauchy sequences

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    An ideal II is a family of subsets of positive integers N\textbf{N} which is closed under taking finite unions and subsets of its elements. A sequence (xn)(x_n) of real numbers is said to be II-convergent to a real number LL, if for each \;ε>0 \varepsilon> 0 the set {n:xnLε}\{n:|x_{n}-L|\geq \varepsilon\} belongs to II. We introduce II-ward compactness of a subset of R\textbf{R}, the set of real numbers, and II-ward continuity of a real function in the senses that a subset EE of R\textbf{R} is II-ward compact if any sequence (xn)(x_{n}) of points in EE has an II-quasi-Cauchy subsequence, and a real function is II-ward continuous if it preserves II-quasi-Cauchy sequences where a sequence (xn)(x_{n}) is called to be II-quasi-Cauchy when (Δxn)(\Delta x_{n}) is II-convergent to 0. We obtain results related to II-ward continuity, II-ward compactness, ward continuity, ward compactness, ordinary compactness, ordinary continuity, δ\delta-ward continuity, and slowly oscillating continuity.Comment: 16 pages. arXiv admin note: text overlap with arXiv:1005.494