Ideal-quasi-Cauchy sequences

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.Comment: 16 pages. arXiv admin note: text overlap with arXiv:1005.494

Upward and downward statistical continuities

A real valued function $f$ defined on a subset $E$ of $\textbf{R}$, the set of real numbers, is statistically upward continuous if it preserves statistically upward half quasi-Cauchy sequences, is statistically downward continuous if it preserves statistically downward half quasi-Cauchy sequences; and a subset $E$ of $\textbf{R}$, is statistically upward compact if any sequence of points in $E$ has a statistically upward half quasi-Cauchy subsequence, is statistically downward compact if any sequence of points in $E$ has a statistically downward half quasi-Cauchy subsequence where a sequence $(x_{n})$ of points in $\textbf{R}$ is called statistically upward half quasi-Cauchy if $$\lim_{n\rightarrow\infty}\frac{1}{n}|\{k\leq n: x_{k}-x_{k+1}\geq \varepsilon\}|=0$$ is statistically downward half quasi-Cauchy if $$\lim_{n\rightarrow\infty}\frac{1}{n}|\{k\leq n: x_{k+1}-x_{k}\geq \varepsilon\}|=0$$ for every $\varepsilon>0$. We investigate statistically upward continuity, statistically downward continuity, statistically upward half compactness, statistically downward half compactness and prove interesting theorems. It turns out that uniform limit of a sequence of statistically upward continuous functions is statistically upward continuous, and uniform limit of a sequence of statistically downward continuous functions is statistically downward continuous.Comment: 25 pages. arXiv admin note: substantial text overlap with arXiv:1205.3674, arXiv:1103.1230, arXiv:1102.1531, arXiv:1305.069

On ideal-ward compactness

A family of sets $I\subset2^{\textbf{N}}$ is called an ideal if and only if for each $A,B\in I,$ implies $A\cup B\in I$ and for each $A\in I$ and each $B\subset A,$ implies $B\in I.$ A real function $f$ is ward continuous if and only if $(\Delta f(x_{n}))$ is a null sequence whenever $(x_{n})$ is a null sequence and a subset $E$ of $\textbf{R}$ is ward compact if any sequence $\textbf{x}=(x_{n})$ of points in $E$ has a quasi-Cauchy subsequence where $\textbf{R}$ is the set of real numbers. These recent known results suggest to us introducing a concept of $I$-ward continuity in the sense that a function $f$ is $I$-ward continuous if $I-\lim_{n\rightarrow\infty} \Delta f(x_{n})=0$ whenever $I-\lim_{n\rightarrow\infty} \Delta x_{n}=0$ and a concept of $I$-ward compactness in the sense that a subset $E$ of $\textbf{R}$ is $I$-ward compact if any sequence $\textbf{x}=(x_{n})$ of points in $E$ has a subsequence $\textbf{z}=(z_{k})=(x_{n_{k}})$ of the sequence $\textbf{x}$ such that $I-\lim_{k\rightarrow \infty} \Delta z_{k}=0$ where $\Delta z_{k}=z_{k+1}-z_{k}$. We investigate $I$-ward continuity and $I$-ward compactness, and prove some related problem