# Nθ-Ward Continuity

Authors: Huseyin Cakalli
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

A functionf is continuous if and only if f preserves convergent sequences; that is, (f(αn)) is a convergent sequence whenever (αn) is convergent. The concept of Nθ-ward continuity is defined in the sense that a function f is Nθ-ward continuous if it preserves Nθ-quasi-Cauchy sequences; that is, (f(αn)) is an Nθ-quasi-Cauchy sequence whenever (αn) is Nθ-quasi-Cauchy. A sequence (αk) of points in R, the set of real numbers, is Nθ-quasi-Cauchy if limr→∞(1/hr)∑k∈Ir|Δαk|=0, where Δαk=αk+1-αk, Ir=(kr-1,kr], and θ=(kr) is a lacunary sequence, that is, an increasing sequence of positive integers such that k0=0 and hr:kr-kr-1→∞. A new type compactness, namely, Nθ-ward compactness, is also, defined and some new results related to this kind of compactness are obtained.


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## Body

## 1. Introduction

It is well known that a real functionf is continuous if and only if, for each point α0 in the domain, limn→∞f(αn)=f(α0) whenever limn→∞αn=α0. This is equivalent to the statement that (f(αn)) is a convergent sequence whenever (αn) is. This is also equivalent to the statement that (f(αn)) is a Cauchy sequence whenever (αn) is Cauchy provided that the domain of the function is either whole R or a bounded and closed subset of R where R is the set of real numbers. These well known results for continuity for real functions in terms of sequences suggested to introduce and study new types of continuities such as slowly oscillating continuity [1], quasi-slowly oscillating continuity [2], δ-quasi-slowly oscillating continuity [3], forward continuity [4], statistical ward continuity [5] which enabled some authors to obtain some characterizations of uniform continuity in terms of sequences in the sense that a function preserves either quasi-Cauchy sequences or slowly oscillating sequences (see [6–8]).The purpose of this paper is to introduce a new kind of continuity and a new type of compactness, namely,Nθ-ward continuity and Nθ-ward compactness, respectively, in the senses that a function f is Nθ-ward continuous if f preserves Nθ-quasi-Cauchy sequences, and a subset A of R is Nθ-ward compact if any sequence of points in A has an Nθ-quasi-Cauchy subsequence and to investigate relations among this kind of continuity, compactness, and some other types of continuities.


## 2. Preliminaries

We will use boldface lettersα,x,y,z,… for sequences α=(αk),x=(xn),y=(yn), and z=(zn),… of points in R for the sake of abbreviation. s and c will denote the set of all sequences and the set of convergent sequences of points in R.A subset ofR is compact if and only if it is closed and bounded. A subset A of R is bounded if |a|≤M for all a∈A where M is a positive real constant number. This is equivalent to the statement that any sequence of points in A has a Cauchy subsequence. The concept of a Cauchy sequence involves far more than that the distance between successive terms is tending to zero. Nevertheless, sequences which satisfy this weaker property are interesting in their own right. A sequence (αn) of points in R is quasi-Cauchy if (Δαn) is a null sequence where Δαn=αn+1-αn. These sequences were named as quasi-Cauchy by Burton and Coleman [8, page 328], while they were called as forward convergent to 0 sequences in [4, page 226].It is known that a sequence(αn) of points in R is slowly oscillating if(2.1)lim⁡λ→1+lim⁡n¯max⁡n+1≤k≤[λn]|αk-αn|=0,
where [λn] denotes the integer part of λn (see [9, Definition 2 page 947]). Any Cauchy sequence is slowly oscillating, and any slowly oscillating sequence is quasi-Cauchy. There are quasi-Cauchy sequences which are not Cauchy. For example, the sequence (n) is quasi-Cauchy, but not Cauchy. Any subsequence of a Cauchy sequence is Cauchy. The analogous property fails for quasi-Cauchy sequences, and fails for slowly oscillating sequences as well. A counter example for the case, quasi-Cauchy, is again the sequence (an)=(n) with the subsequence (an2)=(n). A counter example for the case slowly oscillating is the sequence (log10n) with the subsequence (n). Furthermore we give more examples without neglecting: the sequences (∑k=1∞1/n),(lnn),(ln(lnn)),(ln(ln(lnn))),…,(ln(ln(ln(⋯(lnn)⋯)))) and combinations like that are all slowly oscillating, but not Cauchy. The bounded sequence (cos(6log(n+1))) is slowly oscillating, but not Cauchy. The sequences (cos(πn)) and (∑k=1k=n(1/k)(∑j=1j=k(1/j))) are quasi-Cauchy, but not slowly oscillating.By a method of sequential convergence, or briefly a method, we mean a linear functionG defined on a subspace of s, denoted by cG, into R. A sequence x=(xn) is said to be G-convergent to ℓ if α∈cG and G(α)=ℓ [10]. In particular, lim denotes the limit function limα=limnαn on the space c of convergent sequences of points in R. A method G is called regular if c⊂cG; that is, every convergent sequence α=(αn) is G-convergent with G(α)=limα. A point ℓ in R is in the G-sequential closure of a subset A of R if there is a sequence x=(xn) of points in A such that G(x)=ℓ. A subset A is called G-sequentially closed if it contains all of the points in its G-sequential closure.Consider an infinite matrixA=(ank)n,k=1∞ of real numbers. Then, for any sequence x=(xn) the sequence Ax is defined as(2.2)Ax=(∑k=1∞ankxk)nprovided that each of the series converges. A sequencex is called A-convergent (or A-summable) to ℓ if Ax exists and is convergent with(2.3)lim⁡Ax=lim⁡n→∞∑k=1∞ankxk=l.Thenℓ is called the A-limit of x. We have thus defined a method of sequential convergence, that is, G(x)=limAx, called a matrix method or a summability matrix.The concept of statistical convergence is a generalization of the usual notion of convergence that, for real-valued sequences, parallels the usual theory of convergence. A sequence(αk) of points in R is called statistically convergent to an element ℓ of R if for each ɛ(2.4)lim⁡n→∞1n|{k≤n:|αk-l|≥ɛ}|=0,and this is denoted by st-limk→∞αk=ℓ (see [11–15]). This defines a method of sequential convergence, that is, G(α):=st-limk→∞αk.Now we recall the concepts of ward compactness, and slowly oscillating compactness: a subsetA of R is called ward compact if whenever (αn) is a sequence of points in A, there is a quasi-Cauchy subsequence z=(zk)=(αnk) of (αn) [4]. A subset A of R is called slowly oscillating compact if whenever (αn) is a sequence of points in A, there is a slowly oscillating subsequence z=(zk)=(αnk) of (αn) [1].A functionf is called G-sequentially continuous at u∈R if, given a sequence α=(αn) of points in R, G(α)=u implies that G(f(α))=f(u).Recently, Cakalli (see [16, page 594], [17]) gave a sequential definition of compactness, which is a generalization of ordinary sequential compactness, as in the following: a subset A of R is G-sequentially compact if for any sequence (αk) of points in A there exists a subsequence z of the sequence such that G(z)∈A. His idea enables us obtaining new kinds of compactness via most of the nonmatrix sequential convergence methods as well as all matrix sequential convergence methods.


## 3.Nθ-Quasi-Cauchy Sequences

A lacunary sequenceθ=(kr) is an increasing sequence θ=(kr) of positive integers such that k0=0 and hr:kr-kr-1→∞. The intervals determined by θ will be denoted by Ir=(kr-1,kr], and the ratio kr/kr-1 will be abbreviated by qr. Sums of the form ∑kr-1+1kr|αk| frequently occur, and will often be written for convenience as ∑k∈Ir|αk|. Throughout this paper, we will assume that liminfrqr>1.The notion ofNθ convergence was introduced and studied by Freedman et al. in [18]. Basarir and Altundag studied Δ-Nθ-asymptotically equivalent sequences in [19]. Using the idea of Sember and Raphael, Fridy and Orhan introduced lacunary statistical convergence (see [20, 21]).A sequence(αk) of points in R is called Nθ-convergent to an element ℓ of R if(3.1)lim⁡r→∞1hr∑k∈Ir|αk-l|=0,and it is denoted byNθ-limαk=ℓ. This defines a method of sequential convergence, that is, G(α):=Nθ-limαk. Any convergent sequence is Nθ-convergent, but the converse is not always true. Throughout the paper Nθ will denote the set of Nθ convergent sequences of points in R.For example, limit of the sequence of the ratios of Fibonacci numbers converge, to the golden mean. This property ensures the regularity of lacunary sequential method obtained via the sequence of Fibonacci numbers; that is,θ=(kr) is the lacunary sequence defined by writing k0=0 and kr=Fr+2 where (Fr) is the Fibonacci sequence, that is, F1=1, F2=1, and Fr=Fr-1+Fr-2 for r≥3.Now we modify the definition ofG-sequential compactness to the special case, G=Nθ [16] as in the following: a subset A, of R is called Nθ-sequentially compact if whenever (αn) is a sequence of points in A there is an Nθ-convergent subsequence z=(zk)=(ank) of (αn) whose Nθ-limit is in A.Adopting the technique in the proof of the necessity of  Theorem  6 in [22], we see that the sequential method Nθ is subsequential. It follows from [16, Corollary 5, page 597] that a subset A of R is sequentially compact if and only if it is Nθ-sequentially compact. A subset A of R is closed and bounded if and only if it is Nθ-sequentially compact. A subset of R is G-sequentially compact if and only if it is Nθ-sequentially compact for any regular subsequential method G.In connection withNθ-convergent sequences and convergent sequences the problem arises to investigate the following types of continuity of functions on R:(3.2)(Nθ):(αn)∈Nθ⟹(f(αn))∈Nθ,(Nθc):(αn)∈Nθ⟹(f(αn))∈c,(c):(αn)∈c⟹(f(αn))∈c,(cNθ):(αn)∈c⟹(f(αn))∈Nθ.We see that(Nθ) is Nθ-sequential continuity of f, and (c) is the ordinary continuity of f. It is easy to see that (Nθc) implies (Nθ), and (Nθ) does not imply (Nθc);  (Nθ) implies (cNθ), and (cNθ) does not imply (Nθ); (Nθc) implies (c), and (c) does not imply (Nθc); and (c) is equivalent to (cNθ).If a functionf is Nθ-sequentially continuous at a point α0, then it is continuous at α0. If a function f is Nθ-sequentially continuous on a subset A of R, then it is statistically continuous on A. We obtain from [16, Theorem 7, page 597] that Nθ-sequentially continuous image of any Nθ-sequentially compact subset of R is Nθ-sequentially compact.In [23] a nonempty subset A of a R is called G-sequentially connected if there are no nonempty and disjoint G-sequentially closed subsets U and V such that A⊆U⋃V, and A∩U and A∩V are nonempty. As far as G-sequentially connectedness is considered, we see that Nθ-sequentially continuous image of any Nθ-sequentially connected subset of R is Nθ-sequentially connected, so Nθ-sequentially continuous image of any interval is an interval. Furthermore it can be easily seen that a subset of R is Nθ-sequentially connected if and only if it is connected in the ordinary sense, and so it is an interval.Definition 3.1.
A sequence(αn) of points in R is called Nθ-quasi-Cauchy if (Δαn) is Nθ-convergent to 0. ΔNθ0 will denote the set of all Nθ-quasi-Cauchy sequences of points in R.
We note thatNθ-quasi-Cauchy sequences were studied in [19] in a different point of view.Now we give the definition ofNθ-ward compactness.Definition 3.2.
A subsetA of R is called Nθ-ward compact if whenever (αn) is a sequence of points in A, there is an Nθ-quasi-Cauchy subsequence z=(zk)=(ank) of (αn).Theorem 3.3.
A subsetA of R is bounded if and only if it is Nθ-ward compact.Proof.
LetA be any bounded subset of R and let (αn) be any sequence of points in A. (αn) is also a sequence of points in A¯ where A¯ denotes the closure of A. As A¯ is sequentially compact, there is a convergent subsequence (αnk) of (αn) (no matter the limit is in A or not). This subsequence is Nθ-convergent since Nθ-method is regular. Hence (αnk) is Nθ-quasi-Cauchy. Thus (a) implies (b). To prove that (b) implies (a), suppose that A is unbounded. If it is unbounded above, then one can construct a sequence (αn) of numbers in A such that αn+1>1+αn for each positive integer n. Then the sequence (αn) does not have any Nθ-quasi-Cauchy subsequence, so A is not Nθ-ward compact. If A is bounded above and unbounded below, then similarly we obtain that A is not Nθ-ward compact. This completes the proof of the theorem.It easily follows from the preceding theorem that a closed subset ofR is Nθ-ward compact if and only if it is Nθ-sequentially compact and a closed subset of R is Nθ-ward compact if and only if it is statistically ward compact.A sequenceα=(αn) is δ-quasi-Cauchy if limk→∞Δ2αn=0 where Δ2αn=an+2-2an+1+αn [3]. A subset A of R is called δ-ward compact if whenever α=(αn) is a sequence of points in A, there is a subsequence z=(zk)=(αnk) of α with limk→∞Δ2zk=0. It follows from the previous theorem that any Nθ-ward compact subset of R is δ-ward compact.We see that for any regular subsequential methodG defined on R, if a subset A of R is G-sequentially compact, then it is Nθ-ward compact. But the converse is not always true.Now we give the definition ofNθ-ward continuity in the following.Definition 3.4.
A function defined on a subsetA of R is called Nθ-ward continuous if it preserves Nθ-quasi-Cauchy sequences; that is, (f(αk)) is an Nθ-quasi-Cauchy sequence whenever (αk) is.Sum of twoNθ-ward continuous functions is Nθ-ward continuous, but product of Nθ-ward continuous functions need not be Nθ-ward continuous.In connection withNθ-quasi-Cauchy sequences and convergent sequences the problem arises to investigate the following types of continuity of functions on R:(3.3)(δNθ):(αn)∈ΔNθ0⟹(f(αn))∈ΔNθ0,(δNθc):(αn)∈ΔNθ0⟹(f(αn))∈c,(c):(αn)∈c⟹(f(αn))∈c,(cδNθ):(αn)∈c⟹(f(αn))∈ΔNθ0,(Nθ):(αn)∈Nθ⟹(f(αn))∈Nθ0.We see that(δNθ) is Nθ-ward continuity of f, (Nθ) is Nθ-sequential continuity of f, and (c) is the ordinary continuity of f. It is easy to see that (δNθc) implies (δNθ), and (δNθ) does not imply (δNθc); (δNθ) implies (cδNθ), and (cδNθ) does not imply (δNθ); (δNθc) implies (c), and (c) does not imply (δNθc); (Nθ) clearly implies (c) as we have seen in Section 3.Now we give the implication that(δNθ) implies (Nθ); that is, any Nθ-ward continuous function is Nθ-sequentially continuous.Theorem 3.5.
Iff is Nθ-ward continuous on a subset A of R, then it is Nθ-sequentially continuous on A.Proof.
Assume thatf is an Nθ-ward continuous function on a subset A of R. Let (αn) be any Nθ-convergent sequence with Nθ-limk→∞αk=α0. Then the sequence
(3.4)(α1,α0,α2,α0,…,αn-1,α0,αn,α0,…)
is also Nθ-convergent to α0. Hence it is Nθ-quasi-Cauchy. As f is Nθ-ward continuous, the sequence
(3.5)(f(α1),f(α0),f(α2),f(α0),…,f(αn-1),f(α0),f(αn),f(α0),…)
is Nθ-quasi-Cauchy. It follows from this that the sequence (f(αn))Nθ-converges to f(α0). This completes the proof of the theorem.
The converse is not always true for the functionf(x)=x2 is an example since the sequence (n) is Nθ-quasi-Cauchy while (f(n))=(n) is not.Corollary 3.6.
Iff is Nθ-ward continuous on a subset A of R, then it is continuous on A.Proof.
The proof immediately follows from the preceding theorem, so it is omitted.Corollary 3.7.
Iff is Nθ-ward continuous on a subset A of R, then it is statistically continuous on A.It is well known that any continuous function on a compact subsetA of R is uniformly continuous on A. It is also true for a regular subsequential method G that any Nθ-ward continuous function on a G-sequentially compact subset A of R is also uniformly continuous on A (see [6]). Furthermore, for Nθ-ward continuous functions defined on an Nθ-ward compact subset of R, we have the following.Theorem 3.8.
LetA be an Nθ-ward compact subset A of R and let f:A→R be an Nθ-ward continuous function on A. Then f is uniformly continuous on A.Proof.
Suppose thatf is not uniformly continuous on A so that there exists an ɛ0>0 such that for any δ>0 there are x,y∈E with |x-y|<δ but |f(x)-f(y)|≥ɛ0. For each positive integer n, there exist αn and βn such that |αn-βn|<1/n, and |f(αn)-f(βn)|≥ɛ0. Since A is Nθ-ward compact, there exists an Nθ-quasi-Cauchy subsequence (αnk) of the sequence (αn). It is clear that the corresponding subsequence (βnk) of the sequence (βn) is also Nθ-quasi-Cauchy, since (βnk+1-βnk) is a sum of three Nθ-null sequences, that is,
(3.6)βnk+1-βnk=(βnk+1-αnk+1)+(αnk+1-αnk)+(αnk-βnk).
On the other hand, it follows from the equality αnk+1-βnk=αnk+1-αnk+αnk-βnk that the sequence (αnk+1-βnk) is Nθ-convergent to 0. Hence the sequence
(3.7)(an1,βn1,αn2,βn2,αn3,βn3,…,αnk,βnk,…)
is Nθ-quasi-Cauchy. But the transformed sequence
(3.8)(f(αn1),f(βn1),f(αn2),f(βn2),f(αn3),f(βn3),…,f(αnk),f(βnk),…)
is not Nθ-quasi-Cauchy. Thus f does not preserve Nθ-quasi-Cauchy sequences. This contradiction completes the proof of the theorem.Corollary 3.9.
If a functionf is Nθ-ward continuous on a bounded subset A of R, then it is uniformly continuous on A.Proof.
The proof follows from the preceding theorem and Theorem3.3.Theorem 3.10.
Nθ-ward continuous image of any Nθ-ward compact subset of R is Nθ-ward compact.Proof.
Assume thatf is an Nθ-ward continuous function on a subset A of R and E is an Nθ-ward compact subset of A. Let (βn) be any sequence of points in f(E). Write βn=f(αn) where αn∈E for each positive integer n. Nθ-ward compactness of E implies that there is a subsequence (γk)=(αnk) of (αn) with Nθlimk→∞Δγk=0. Write (tk)=(f(γk)). As f is Nθ-ward continuous, (f(γk)) is Nθ-quasi-Cauchy. Thus we have obtained a subsequence (tk) of the sequence (f(αn)) with Nθ-limk→∞Δtk=0. Thus f(E) is Nθ-ward compact. This completes the proof of the theorem.Corollary 3.11.
Nθ-ward continuous image of any compact subset of R is Nθ-ward compact.The proof follows from the preceding theorem.Corollary 3.12.
Nθ-ward continuous image of any bounded subset of R is bounded.The proof follows from Theorems3.3 and 3.10.Corollary 3.13.
Nθ-ward continuous image of a G-sequentially compact subset of R is Nθ-ward compact for any regular subsequential method G.For a further study, we suggest to investigateNθ-quasi-Cauchy sequences of fuzzy points and Nθ-ward continuity for the fuzzy functions (see [24] for the definitions and related concepts in fuzzy setting). However due to the change in settings, the definitions and methods of proofs will not always be analogous to those of the present work.



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