Half quasi-Cauchy sequences

Mathematical and Computer Modelling, 2011

Recently, it has been proved that a real-valued function defined on a subset E of R, the set of real numbers, is uniformly continuous on E if and only if it is defined on E and preserves quasi-Cauchy sequences of points in E where a sequence is called quasi-Cauchy if ( x n ) is a null sequence. In this paper we call a real-valued function defined on a subset E of R δ-ward continuous if it preserves δ-quasi-Cauchy sequences where a sequence x = (x n ) is defined to be δ-quasi-Cauchy if the sequence ( x n ) is quasi-Cauchy. It turns out that δ-ward continuity implies uniform continuity, but there are uniformly continuous functions which are not δ-ward continuous. A new type of compactness in terms of δ-quasi-Cauchy sequences, namely δ-ward compactness is also introduced, and some theorems related to δ-ward continuity and δ-ward compactness are obtained.

Abstract and Applied Analysis, 2013

Recently, the concept of -ward continuity was introduced and studied. In this paper, we prove that the uniform limit of -ward continuous functions is -ward continuous, and the set of all -ward continuous functions is a closed subset of the set of all continuous functions. We also obtain that a real function defined on an interval is uniformly continuous if and only if (()) is -quasi-Cauchy whenever () is a quasi-Cauchy sequence of points in .

Filomat

In this paper, we investigate the concepts of downward continuity and upward continuity. A real valued function on a subset E of R, the set of real numbers, is downward continuous if it preserves downward quasi-Cauchy sequences; and is upward continuous if it preserves upward quasi-Cauchy sequences, where a sequence (xk) of points in R is called downward quasi-Cauchy if for every ? > 0 there exists an n0 ? N such that xn+1 - xn < ? for n ? n0, and called upward quasi-Cauchy if for every ? > 0 there exists an n1 ? N such that xn - xn+1 < ? for n ? n1. We investigate the notions of downward compactness and upward compactness and prove that downward compactness coincides with above boundedness. It turns out that not only the set of downward continuous functions, but also the set of upward continuous functions is a proper subset of the set of continuous functions.

2021

The underlying theme of this article is a class of sequences in metric structures satisfying a much weaker kind of Cauchy condition, namely quasi-Cauchy sequences (introduced in [7]) that has been used to define several new concepts in recent articles [9, 10]. We first introduce a new notion of precompactness based on the idea of quasi-Cauchy sequences and establish several results including a new characterization of compactness in metric spaces. Next we consider associated idea of continuity, namely, ward continuous functions [8], as this class of functions strictly lies between the classes of continuous and uniformly continuous functions and mainly establish certain coincidence results. Finally a new class of Lipschitz functions called “quasi-Cauchy Lipschitz functions” is introduced following the line of investigations in [3, 4, 5, 12] and again several coincidence results are proved along with a very interesting observation that every real valued ward continuous function defined...

Journal of Inequalities and Applications, 2012

An ideal I is a family of subsets of positive integers N which is closed under taking finite unions and subsets of its elements. A sequence (xn) of real numbers is said to be I-convergent to a real number L, if for each ε > 0 the set {n : |xn − L| ≥ ε} belongs to I. We introduce I-ward compactness of a subset of R, the set of real numbers, and I-ward continuity of a real function in the senses that a subset E of R is I-ward compact if any sequence (xn) 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 (xn) is called to be I-quasi-Cauchy when (∆xn) is I-convergent to 0. We obtain results related to I-ward continuity, I-ward compactness, ward continuity, ward compactness, ordinary compactness, ordinary continuity, δward continuity, and slowly oscillating continuity.

For a fixed positive i nteger p, a sequence (x n) in a metric space X is c alled p-quasi-Cauchy if (Δ p x n) is a null sequence where Δ p x n = d(x n+p , x n) for each positive integer n. A subset E of X is called p-ward compact if any sequence (x n) of points in E has a p-quasi-Cauchy subsequence. A subset of X is totally bounded if and only if it is p-ward compact. A function f from a subset E of X into a metric space Y is called p-ward continuous if it preserves p-quasi Cauchy sequences, i.e. (f (x n)) is a p-quasi Cauchy sequence in Y whenever (x n) is a p-quasi Cauchy sequence of points of E. A function f from a totally bounded subset of X into Y preserves p-quasi Cauchy sequences if and only if it is uniformly continuous. If a function is uniformly continuous on a subset E of X into Y, then (f (x n) is p-quasi Cauchy in Y whenever (x n) is a quasi cauchy sequence of points in E.

Filomat, 2015

In this paper, we introduce and study new kinds of continuities. It turns out that a function f defined on an interval is uniformly continuous if and only if there exists a positive integer p such that f preserves p-quasi-Cauchy sequences where a sequence (xn) is called p-quasi-Cauchy if the sequence of differences between p-successive terms tends to 0.

arXiv: Functional Analysis, 2018

In this paper, we introduce and investigate the concepts of down continuity and down compactness. A real valued function $f$ on a subset $E$ of $\R$, the set of real numbers is down continuous if it preserves downward half Cauchy sequences, i.e. the sequence $(f(\alpha_{n}))$ is downward half Cauchy whenever $(\alpha_{n})$ is a downward half Cauchy sequence of points in $E$, where a sequence $(\alpha_{ k})$ of points in $\R$ is called downward half Cauchy if for every $\varepsilon>0$ there exists an $n_{0}\in{\N}$ such that $\alpha_{m}-\alpha_{n} <\varepsilon$ for $m \geq n \geq n_0$. It turns out that the set of down continuous functions is a proper subset of the set of continuous functions.

Math Comput Modelling, 2011

In this paper we generalize the concept of a quasi-Cauchy sequence to a concept of a p-quasi-Cauchy sequence for any fixed positive integer p. For p = 1 we obtain some earlier existing results as a special case. We obtain some interesting theorems related to p-quasi-Cauchy continuity, G-sequential continuity, slowly oscillating continuity, and uniform continuity. It turns out that a function f defined on an interval is uniformly continuous if and only if there exists a positive integer p such that f preserves p-quasi-Cauchy sequences where a sequence (xn) is called p-quasi-Cauchy if (x n+p − xn) ∞ n=1 is a null sequence. Recently, in [1], a concept of quasi-Cauchy continuity, and a concept of quasi-Cauchy compactness have been introduced in the senses that a real function is called quasi-Cauchy continuous if lim n→∞ ∆f (x n) = 0 whenever lim n→∞ ∆x n = 0, and a subset E of R is called quasi-Cauchy compact if whenever (x n) is a sequence of points in E there is a subsequence (y k) = (x n k) of (x n) with lim k→∞ ∆y k = 0 where ∆y k = y k+1 − y k. We note that forward continuity and forward compactness

2012

In this paper we generalize the concept of a quasi-Cauchy sequence to a concept of a p-quasi-Cauchy sequence for any fixed positive integer p. For p = 1 we obtain some earlier existing results as a special case. We obtain some interesting theorems related to p-quasi-Cauchy continuity, G-sequential continuity, slowly oscillating continuity, and uniform continuity. It turns out that a function f defined on an interval is uniformly continuous if and only if there exists a positive integer p such that f preserves p-quasi-Cauchy sequences where a sequence (xn) is called p-quasi-Cauchy if (x n+p − xn) ∞ n=1 is a null sequence. Recently, in [1], a concept of quasi-Cauchy continuity, and a concept of quasi-Cauchy compactness have been introduced in the senses that a real function is called quasi-Cauchy continuous if lim n→∞ ∆f (x n) = 0 whenever lim n→∞ ∆x n = 0, and a subset E of R is called quasi-Cauchy compact if whenever (x n) is a sequence of points in E there is a subsequence (y k) = (x n k) of (x n) with lim k→∞ ∆y k = 0 where ∆y k = y k+1 − y k. We note that forward continuity and forward compactness