Upward and downward statistical continuities

Mathematical and Computer Modelling, 2011

A subset E of a metric space (X, d) is totally bounded if and only if any sequence of points in E has a Cauchy subsequence. We call a sequence (x n) statistically quasi-Cauchy if st − lim n→∞ d(x n+1 , x n) = 0, and lacunary statistically quasi-Cauchy if S θ − lim n→∞ d(x n+1 , x n) = 0. We prove that a subset E of a metric space is totally bounded if and only if any sequence of points in E has a subsequence which is any type of the following: statistically quasi-Cauchy, lacunary statistically quasi-Cauchy, quasi-Cauchy, and slowly oscillating. It turns out that a function defined on a connected subset E of a metric space is uniformly continuous if and only if it preserves either quasi-Cauchy sequences or slowly oscillating sequences of points in E.

Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics, 2017

In this paper, the concept of a lacunary statistically-quasi-Cauchy sequence is investigated. In this investigation, we proved interesting theorems related to lacunary statistically-ward continuity, and some other kinds of continuities. A real valued function f de…ned on a subset A of R, the set of real numbers, is called lacunary statistically ward continuous on A if it preserves lacunary statistically delta quasi-Cauchy sequences of points in A, i.e. (f (k)) is a lacunary statistically delta quasi-Cauchy sequence whenever (k) is a lacunary statistically delta quasi-Cauchy sequence of points in A, where a sequence (k) is called lacunary statistically delta quasi-Cauchy if (k) is a lacunary statistically quasi-Cauchy sequence. It turns out that the set of lacunary statistically ward continuous functions is a closed subset of the set of continuous functions.

arXiv: Functional Analysis, 2017

In this paper, we introduce a concept of statistically $p$-quasi-Cauchyness of a real sequence in the sense that a sequence $(\alpha_{k})$ is statistically $p$-quasi-Cauchy if $\lim_{n\rightarrow\infty}\frac{1}{n}|\{k\leq n: |\alpha_{k+p}-\alpha_{k}|\geq{\varepsilon}\}|=0$ for each $\varepsilon>0$. A function $f$ is called statistically $p$-ward continuous on a subset $A$ of the set of real umbers $\mathbb{R}$ if it preserves statistically $p$-quasi-Cauchy sequences, i.e. the sequence $f(\textbf{x})=(f(\alpha_{n}))$ is statistically $p$-quasi-Cauchy whenever $\boldsymbol\alpha=(\alpha_{n})$ is a statistically $p$-quasi-Cauchy sequence of points in $A$. It turns out that a real valued function $f$ is uniformly continuous on a bounded subset $A$ of $\mathbb{R}$ if there exists a positive integer $p$ such that $f$ preserves statistically $p$-quasi-Cauchy sequences of points in $A$.

2011

The main object of this paper is to investigate lacunary statistically ward continuity. We obtain some relations between this kind of continuity and some other kinds of continuities. It turns out that any lacunary statistically ward continuous real valued function on a lacunary statistically ward compact subset $E\subset{\textbf{R}}$ is uniformly continuous.

The main object of this paper is to investigate $\lambda$-statistically quasi-Cauchy sequences. A real valued function $f$ defined on a subset $E$ of $\textbf{R}$, the set of real numbers, is called $\lambda$-statistically ward continuous on $E$ if it preserves $\lambda$-statistically quasi-Cauchy sequences of points in $E$. It turns out that uniform continuity coincides with $\lambda$-statistically ward continuity on $\lambda$-statistically ward compact subsets.

2017

A real valued function defined on a subset $E$ of $\mathbb{R}$, the set of real numbers, is $\rho$-statistically downward continuous if it preserves $\rho$-statistical downward quasi-Cauchy sequences of points in $E$, where a sequence $(\alpha_{k})$ of real numbers is called ${\rho}$-statistically downward quasi-Cauchy if $\lim_{n\rightarrow\infty}\frac{1}{\rho_{n} }|\{k\leq n: \Delta \alpha_{k} \geq \varepsilon\}|=0 $ for every $\varepsilon>0$, in which $(\rho_{n})$ is a non-decreasing sequence of positive real numbers tending to $\infty$ such that $\limsup _{n} \frac{\rho_{n}}{n}<\infty $, $\Delta \rho_{n}=O(1)$, and $\Delta \alpha _{k} =\alpha _{k+1} - \alpha _{k}$ for each positive integer $k$. It turns out that a function is uniformly continuous if it is $\rho$-statistical downward continuous on an above bounded set.

Filomat

In this paper, we investigate the concept of Abel statistical quasi Cauchy sequences. A real function f is called Abel statistically ward continuous if it preserves Abel statistical quasi Cauchy sequences, where a sequence (?k) of point in R is called Abel statistically quasi Cauchy if limx?1-(1-x) ?k:|??k|?? xk = 0 for every ? > 0, where ??k = ?k+1-?k for every k ? N. Some other types of continuities are also studied and interesting results are obtained. It turns out that the set of Abel statistical ward continuous functions is a closed subset of the space of continuous functions.

A sequence (α k) of points in R, the set of real numbers, is called ρ-statistically p quasi Cauchy if lim n→∞ 1 ρ n |{k ≤ n : |∆ p α k | ≥ ε}| = 0 for each ε > 0, where ρ = (ρ n) is a non-decreasing sequence of positive real numbers tending to ∞ such that lim sup n ρn n < ∞, ∆ρ n = O(1), and ∆ p α k+p = α k+p − α k for each positive integer k, p is a fixed positive integer. A real-valued function defined on a subset of R is called ρ-statistically p-ward continuous if it preserves ρ-statistical p-quasi Cauchy sequences. We obtain results related to ρ-statistical p-ward continuity, ρ-statistical p-ward compactness, p-ward continuity, ward continuity, and uniform continuity.

This paper is withdrawn because the results in the paper are included in a paper to be published in Mathematical and Computer Modelling.

A function $f$ defined on a subset $E$ of a two normed space $X$ is statistically ward continuous if it preserves statistically quasi-Cauchy sequences of points in $E$ where a sequence $(x_n)$ is statistically quasi-Cauchy if $(\Delta x_{n})$ is a statistically null sequence. A subset $E$ of $X$ is statistically ward compact if any sequence of points in $E$ has a statistically quasi-Cauchy subsequence. In this paper, new kinds of continuities are investigated in two normed spaces. It turns out that uniform limit of statistically ward continuous functions is again statistically ward continuous.

THIRD INTERNATIONAL CONFERENCE OF MATHEMATICAL SCIENCES (ICMS 2019)

In this study, we investigate the concepts of Abel statistical convergence and Abel statistical quasi Cauchy sequences. A function f from a subset E of a metric space X into X is called Abel statistically ward continuous if it preserves Abel statistical quasi Cauchyness, where a sequence (x k) of point in E is called Abel statistically quasi Cauchy if lim x→1 − (1− x) k:d(x k+1 ,x k)≥ε x k = 0 for every ε > 0. Some other types of continuities are also studied and interesting results are obtained.

Applied Mathematics Letters, 2011

Recently, it has been proved that a real-valued function defined on an interval A of R, the set of real numbers, is uniformly continuous on A if and only if it is defined on A and preserves quasi-Cauchy sequences of points in A. In this paper we call a real-valued function statistically ward continuous if it preserves statistical quasi-Cauchy sequences where a sequence (α k) is defined to be statistically quasi-Cauchy if the sequence (α k) is statistically convergent to 0. It turns out that any statistically ward continuous function on a statistically ward compact subset A of R is uniformly continuous on A. We prove theorems related to statistical ward compactness, statistical compactness, continuity, statistical continuity, ward continuity, and uniform continuity.

2016

In this paper, the concept of a lacunary statistically-quasi-Cauchy sequence is investigated. In this investigation, we proved interesting theorems related to lacunary statistically-ward continuity, and some other kinds of continuities. A real valued function f de…ned on a subset A of R, the set of real numbers, is called lacunary statistically ward continuous on A if it preserves lacunary statistically delta quasi-Cauchy sequences of points in A, i.e. (f (k)) is a lacunary statistically delta quasi-Cauchy sequence whenever (k) is a lacunary statistically delta quasi-Cauchy sequence of points in A, where a sequence (k) is called lacunary statistically delta quasi-Cauchy if (k) is a lacunary statistically quasi-Cauchy sequence. It turns out that the set of lacunary statistically ward continuous functions is a closed subset of the set of continuous functions.

A real function $f$ is ward continuous if $f$ preserves quasi-Cauchyness, i.e. $(f(x_{n}))$ is a quasi-Cauchy sequence whenever $(x_{n})$ is quasi-Cauchy; and a subset $E$ of $\textbf{R}$ is quasi-Cauchy 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 known results suggest to us introducing a concept of upward (respectively, downward) half quasi-Cauchy continuity in the sense that a function $f$ is upward (respectively, downward) half quasi-Cauchy continuous if it preserves upward (respectively, downward) half quasi-Cauchy sequences, and a concept of upward (respectively, downward) half quasi-Cauchy compactness in the sense that a subset $E$ of $\textbf{R}$ is upward (respectively, downward) half quasi-Cauchy compact if any sequence of points in $E$ has an upward (respectively, downward) half quasi-Cauchy subsequence. We investigate upward(respectively, downward) half quasi-Cauchy conti...

In this paper, we investigate the concept of Abel statistical delta quasi Cauchy sequences. A real function f is called Abel statistically delta ward continuous it preserves Abel statistical delta quasi Cauchy sequences, where a sequence (α k) of point in R is called Abel statistically delta quasi Cauchy if lim x→1 − (1 − x) k:|Δ 2 α k |≥ε x k = 0 for every ε > 0, where Δ 2 α k = α k+2 − 2α k+1 + α k for every k ∈ N. Some other types of continuities are also studied and interesting results are obtained.