scholarly journals A Study on -Quasi-Cauchy Sequences

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Hüseyin Çakalli ◽  
Huseyin Kaplan
Keyword(s):  
Real Function ◽  
Cauchy Sequence ◽  
Closed Subset ◽  
Continuous Functions ◽  
Uniform Limit ◽  
Cauchy Sequences ◽  
Uniformly Continuous

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 .

scholarly journals Abel statistical quasi Cauchy sequences

Filomat ◽  
2019 ◽  
Vol 33 (2) ◽  
pp. 535-541
Author(s):  
Huseyin Cakalli
Keyword(s):  
Real Function ◽  
Closed Subset ◽  
Continuous Functions ◽  
Space Of Continuous Functions ◽  
Cauchy Sequences

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.

scholarly journals Variations on the strongly lacunary quasi Cauchy sequences

Filomat ◽  
2019 ◽  
Vol 33 (2) ◽  
pp. 493-498
Author(s):  
Huseyin Kaplan
Keyword(s):  
Continuous Function ◽  
Cauchy Sequence ◽  
Continuous Functions ◽  
Ordinary Sense ◽  
Uniform Limit ◽  
Cauchy Sequences

In this paper, we introduce concepts of a strongly lacunary p-quasi-Cauchy sequence and strongly lacunary p-ward continuity. We prove that a subset of R is bounded if and only if it is strongly lacunary p-ward compact. It is obtained that any strongly lacunary p-ward continuous function on a subset A of R is continuous in the ordinary sense. We also prove that the uniform limit of strongly lacunary p-ward continuous functions on a subset A of R is strongly lacunary p-ward continuous.

scholarly journals Variations on statistical quasi Cauchy sequences

2019 ◽  
Vol 38 (3) ◽  
pp. 141-150
Author(s):  
Huseyin Cakalli
Keyword(s):  
Positive Integer ◽  
Cauchy Sequence ◽  
Real Sequence ◽  
Bounded Subset ◽  
Cauchy Sequences ◽  
Uniformly Continuous

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 $A$.

scholarly journals New Type Continuities via Abel Convergence

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Huseyin Cakalli ◽  
Mehmet Albayrak
Keyword(s):  
Closed Subset ◽  
Continuous Functions ◽  
Real Numbers ◽  
Uniform Limit ◽  
New Type ◽  
Convergent Sequences

We investigate the concept of Abel continuity. A functionfdefined on a subset ofℝ, the set of real numbers, is Abel continuous if it preserves Abel convergent sequences. Some other types of continuities are also studied and interesting result is obtained. It turned out that uniform limit of a sequence of Abel continuous functions is Abel continuous and the set of Abel continuous functions is a closed subset of continuous functions.

Uniform Continuity of Continuous Functions on Uniform Spaces

1961 ◽  
Vol 13 ◽  
pp. 657-663 ◽  
Author(s):  
Masahiko Atsuji
Keyword(s):  
Real Function ◽  
Metric Spaces ◽  
Uniform Space ◽  
Continuous Functions ◽  
Uniform Continuity ◽  
Uniform Structure ◽  
Uniform Spaces ◽  
Continuous Real Function ◽  
Complete Space ◽  
Uniformly Continuous

Recently several topologists have called attention to the uniform structures (in most cases, the coarsest ones) under which every continuous real function is uniformly continuous (let us call the structures the [coarsest] uc-structures), and some important results have been found which closely relate, explicitly or implicitly, to the uc-structures, such as in the vS of Hewitt (3) and in the e-complete space of Shirota (7). Under these circumstances it will be natural to pose, as Hitotumatu did (4), the problem: which are the uniform spaces with the uc-structures? In (1 ; 2), we characterized the metric spaces with such structures, and in this paper we shall give a solution to the problem in uniform spaces (§ 1), together with some of its applications to normal uniform spaces and to the products of metric spaces (§ 2). It is evident that every continuous real function on a uniform space is uniformly continuous if and only if the uniform structure of the space is finer than the uniform structure defined by all continuous real functions on the space.

scholarly journals Lattices of uniformly continuous functions

2013 ◽  
Vol 160 (1) ◽  
pp. 50-55 ◽  
Author(s):  
Félix Cabello Sánchez ◽  
Javier Cabello Sánchez
Keyword(s):  
Continuous Functions ◽  
Uniformly Continuous

scholarly journals Strongly Lacunary Ward Continuity in 2-Normed Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Hüseyin Çakalli ◽  
Sibel Ersan
Keyword(s):  
Normed Space ◽  
Cauchy Sequence ◽  
Normed Spaces ◽  
Cauchy Sequences

A functionfdefined on a subsetEof a 2-normed spaceXis strongly lacunary ward continuous if it preserves strongly lacunary quasi-Cauchy sequences of points inE; that is,(f(xk))is a strongly lacunary quasi-Cauchy sequence whenever (xk) is strongly lacunary quasi-Cauchy. In this paper, not only strongly lacunary ward continuity, but also some other kinds of continuities are investigated in 2-normed spaces.

Matrix Transformations in an Incomplete Space

1968 ◽  
Vol 20 ◽  
pp. 727-734 ◽  
Author(s):  
I. J. Maddox
Keyword(s):  
Linear Space ◽  
Cauchy Sequence ◽  
Convergent Sequence ◽  
Convergent Series ◽  
Matrix Transformations ◽  
Cauchy Sequences ◽  
Complex Linear ◽  
Incomplete Space ◽  
Bounded Sequences ◽  
Convergent Sequences

Let X = (X, p) be a seminormed complex linear space with zero θ. Natural definitions of convergent sequence, Cauchy sequence, absolutely convergent series, etc., can be given in terms of the seminorm p. Let us write C = C(X) for the set of all convergent sequences for the set of Cauchy sequences; and L∞ for the set of all bounded sequences.

scholarly journals The Bi-dimensional space of Korenblum and composition operator

2015 ◽  
Vol 62 (1) ◽  
pp. 1-12
Author(s):  
José A. Guerrero ◽  
Nelson Merentes ◽  
José L. Sánchez
Keyword(s):  
Banach Space ◽  
Composition Operator ◽  
Real Function ◽  
Dimensional Space ◽  
Similar Type ◽  
Functions Of Two Variables ◽  
Real Functions ◽  
Uniformly Continuous ◽  
Uniformly Bounded

Abstract In this paper we present the concept of total κ-variation in the sense of Hardy-Vitali-Korenblum for a real function defined in the rectangle Iab⊂R2. We show that the space κBV(Iab, R) of real functions of two variables with finite total κ-variation is a Banach space endowed with the norm ||f||κ = |f (a)| + κTV( f, Iab). Also, we characterize the Nemytskij composition operator H that maps the space of functions of two real variables of bounded κ-variation κBV(Iab, R) into another space of a similar type and is uniformly bounded (or Lipschitzian or uniformly continuous).

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