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

1. Introduction

The concept of continuity and any concept involving continuity play a very important role not only in pure mathematics but also in other branches of sciences involving mathematics especially in computer science, information theory, and biological science.

A real function f is continuous if and only if it preserves convergent sequences. A subset E of R, the set of real numbers, is compact if any sequence of points in E has a convergent subsequence whose limit is in E. Using the idea of continuity of a real function and the idea of compactness in terms of sequences, many kinds of continuities and compactness were introduced and investigated, not all but some of them we recall in the following: slowly oscillating continuity, slowly oscillating compactness [1], quasi-slowly oscillating continuity, quasi-slowly oscillating compactness [2], Δ-quasi-slowly oscillating continuity, Δ-quasi-slowly oscillating compactness [3–5], ward continuity, ward compactness [6, 7], δ-ward continuity, δ-ward compactness [8], statistical ward continuity, and lacunary statistical ward continuity [9, 10].

In [11], the notion of Nθ convergence was introduced, and studied by Freedman et al. Using the main idea for continuity and compactness given above the concepts of Nθ-ward compactness of a subset E of R and Nθ-ward continuity of a real function are introduced and investigated recently in [12].

The purpose of this paper is to continue the investigation given in [12] and obtain further interesting results on Nθ-ward continuity.

2. Preliminaries

Boldface letters α, x, y, z,… will be used for sequences α=(αk), α=(xn), y=(yn), z=(zn),… of points in the set of real numbers R for the sake of abbreviation. Sums of the form ∑kr-1+1kr|αk| frequently occur and will often be written for convenience as ∑k∈Ir|αk|.

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 [13, page 328], while they were called as forward convergent to 0 sequences in [7, page 226]. A sequence (αk) of points in R is called Nθ-convergent to an element L of R if
(1)limr→∞1hr∑k∈Ir|αk-L|=0,
where 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→∞. The intervals determined by θ are denoted by Ir=(kr-1,kr], and the ratio kr/kr-1 is abbreviated by qr. A sequence (αn) of points in R is called Nθ-quasi-Cauchy if (Δαn) is Nθ-convergent to 0. A function defined on a subset A 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. |ΔNθ0| will denote the set of Nθ-quasi-Cauchy sequences of points in R. Any subsequence of a Cauchy sequence is Cauchy. The analogous property fails for Nθ-quasi-Cauchy sequences. A counterexample for the case is the sequence (an)=(n) with the subsequence (an2)=(n).

A sequence (αn) of points in R is slowly oscillating [14, Definition 2, page 947] if
(2)limλ→1+limn-maxn+1≤k≤[λn]|αk-αn|=0,
where [λn] denotes the integer part of λn (see also [14]).

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 (αn) of real numbers is said to be ideal convergent to a real number L, if for each ɛ>0 the set {n:|αn-L|≥ɛ} belongs to I. Ideal ward compactness of a subset of R and ideal ward continuity of a real function were recently introduced by Çakalli and Hazarika in [15].

3. Results

Any quasi-Cauchy sequence is Nθ-quasi-Cauchy, so any slowly oscillating sequence is Nθ-quasi-Cauchy, and so any Cauchy sequence is. A sequence α=(αk) is called Cesaro summable to a real number L if limn→∞(1/n)∑i=1nαi=L. This is denoted by C1-limαk=L, and the set of all Cesaro sequences is denoted by σ1. We call a sequence α=(αk) Cesaro quasi-Cauchy if C1-limΔαk=0. The set of all Cesaro quasi-Cauchy sequences is denoted by Δσ10. A sequence α=(αk) is called strongly Cesaro summable to a real number L if limn→∞(1/n)∑i=1n|αi-L|=0. This is denoted by |C1|-limαk=L. The set of all strongly Cesaro summable sequences is denoted by |σ1|. We call a sequence α=(αk) strongly Cesaro quasi-Cauchy if |C1|-limΔαk=0. The set of all strongly Cesaro quasi-Cauchy sequences is denoted by |Δσ10|. The following inclusions are satisfied: |σ1|⊂σ1 and |Δσ10|⊂Δσ10.

Using a similar idea to that of [11], one can easily find out the following inclusion properties between the set of strongly Cesaro quasi-Cauchy sequences and the set of Nθ-quasi-Cauchy sequences (see also [16]).

|ΔNθ0|⊂|Δσ10| if and only if limsupqr<∞ for any lacunary sequence θ.

|Δσ10|⊂|ΔNθ0| if and only if liminfqr>1 for any lacunary sequence θ.

Combining these facts, for any lacunary sequence θ, we have the following:

|ΔNθ0|=|Δσ10| if and only if 1<liminfqr≤limsupqr<∞;

|ΔNθ0|=|Δσ10| if and only if |σ1|=Nθ0.

In the sequel, we will always assume that liminfrqr>1.

We observe that Nθ-summability is a kind of strong A-summability where A=(ark) is a regular matrix generated by the lacunary sequence θ=(kr) as follows:
(3)ark=1hrifk∈Ir,ark=0otherwise.
On the other hand, we see that Nθ-ward continuity cannot be given as a strong A-continuity by any kind of regular summability matrix (related to continuity for strong matrix methods see [17]).

As far as ideal continuity is considered, we note that any Nθ-ward continuous function is ideal continuous; furthermore any Nθ continuous function is ideal continuous for an admissible ideal.

Theorem 1.

If a function f is uniformly continuous on a subset E of R, then, (f(αk)) is Nθ-quasi-Cauchy whenever (αk) is a quasi-Cauchy sequence of points in E.

Proof.

Let E be a subset of R, and let f be a uniformly continuous function on E. Take any quasi-Cauchy sequence (αk) of points in E, and let ɛ be any positive real number. By uniform continuity of f, there exists a δ>0 such that |f(α)-f(β)|<ɛ whenever |α-β|<δ and α,β∈E. Since (αk) is a quasi-Cauchy sequence, there exists a positive integer k0 such that |αk+1-αk|<δ for k≥k0. Hence,
(4)1hr∑k∈Ir|f(αk+1)-f(αk)|<1hr(kr-kr-1)ɛ=ɛ,
for r≥k0. Thus, (f(αk)) is an Nθ-quasi-Cauchy sequence. This completes the proof of the theorem.

We have much more below for a real function f defined on an interval that f is uniformly continuous if and only if (f(αk)) is Nθ-quasi-Cauchy whenever (αk) is a quasi-Cauchy sequence of points in E. First, we give the following lemma.

Lemma 2.

If (ξn,ηn) is a sequence of ordered pairs of points in an interval such that limn→∞|ξn-ηn|=0, then, there exists an Nθ-quasi-Cauchy sequence (αn) with the property that for any positive integer i there exists a positive integer j such that (ξi,ηi)=(αj-1,αj).

Proof.

Although the following proof is similar to that of [13], we give it for completeness. For each positive integer k, we can fix z0k,z1k,…,znkk in E with z0k=ηk, znkk=ξk+1, and |zik-zi-1k|<1/k for 1≤i≤nk. Now write
(5)(ξ1,η1,z11,…,zn1-11,ξ2,η2,z12,…,zn2-12,ξ3,η3,…,ξk,ηk,z1k,…,znk-1k,ξk+1,ηk+1,…).
Then denoting this sequence by (αn), we obtain that for any positive integer i there exists a positive integer j such that (ξi,ηi)=(αj-1,αj). The sequence constructed is a quasi-Cauchy sequence, and it is an Nθ-quasi-Cauchy sequence, since any quasi-Cauchy sequence is an Nθ-quasi-Cauchy sequence. This completes the proof of the lemma.

Theorem 3.

If a function f defined on an interval E is Nθ-ward continuous, then, it is uniformly continuous.

Proof.

Suppose that f is not uniformly continuous on E. Then, there is an ɛ0>0 such that for any δ>0 there exist x,y∈E with |x-y|<δ but |f(x)-f(y)|≥ɛ0. For every integer n≥1 fix ξn, ηn∈E with |ξn-ηn|<1/n and |f(ξn)-f(ηn)|≥ɛ0. By the lemma, there exists an Nθ-quasi-Cauchy sequence (αi) such that for any integer i≥1 there exists a j with ξi=αj and ηi=αj+1. This implies that |f(αj+1)-f(αj)|≥ɛ0; hence, (f(αi)) is not Nθ-quasi-Cauchy. Thus, f does not preserve Nθ-quasi-Cauchy sequences. This completes the proof of the theorem.

Observing that the sequence, constructed in the proof of the preceding theorem, is also a quasi-Cauchy sequence, we obtain that a real function f defined on an interval E is uniformly continuous if (f(αk)) is Nθ-quasi-Cauchy whenever (αk) is a quasi-Cauchy sequence of points in E. Combining this with Theorem 1, we have that a real function f defined on an interval E is uniformly continuous if and only if (f(αk)) is Nθ-quasi-Cauchy whenever (αk) is a quasi-Cauchy sequence of points in E.

Corollary 4.

If a function defined on an interval is Nθ-ward continuous, then, it is ward continuous.

Proof.

The proof follows from Theorem 3 and [7, Theorem 6] so it is omitted.

Corollary 5.

If a function defined on an interval is Nθ-ward continuous, then, it is slowly oscillating continuous.

Proof.

The proof follows from Theorem 3 and [7, Theorem 5] so it is omitted.

It is a well-known result that uniform limit of a sequence of continuous functions is continuous. This is also true in case of Nθ-ward continuity; that is, uniform limit of a sequence of Nθ-ward continuous functions is Nθ-ward continuous.

Theorem 6.

If (fn) is a sequence of Nθ-ward continuous functions on a subset E of R and (fn) is uniformly convergent to a function f, then, f is Nθ-ward continuous on E.

Proof.

Let (αk) be any Nθ-quasi-Cauchy sequence of points in E, and let ɛ be any positive real number. By uniform convergence of (fn), there exists an n1∈N such that |f(α)-fk(α)|<ɛ/3 for n≥n1 and every α∈E. Hence,
(6)1hr∑k∈Ir|f(α)-fk(α)|<1hr(kr-kr-1)ɛ3=ɛ3,
for r≥n1 and every α∈E. As fn1 is Nθ-ward continuous on E, there exists an n2∈N such that for r≥n2(7)1hr∑k∈Ir|fn1(αk+1)-fn1(αk)|<ɛ3.
Now write n0=max{n1,n2}. Thus for r≥n0, we have
(8)1hr∑k∈Ir|f(αk+1)-f(αk)|≤1hr∑k∈Ir|f(αk+1)-fn1(αk+1)|+1hr∑k∈Ir|fn1(αk+1)-fn1(αk)|+1hr∑k∈Ir|fn1(αk)-f(αk)|<ɛ3+ɛ3+ɛ3=ɛ.
Hence,
(9)limr→∞1hr∑k∈Ir|f(αk+1)-f(αk)|=0.
Thus, f preserves Nθ-quasi-Cauchy sequences. This completes the proof of the theorem.

Theorem 7.

The set of all Nθ-ward continuous functions on a subset E of R is a closed subset of the set of all continuous functions on E, that is, ΔNθWC(E)-=ΔNθWC(E), where ΔNθWC(E) is the set of all Nθ-ward continuous functions on E and ΔNθWC(E)- denotes the set of all cluster points of ΔNθWC(E).

Proof.

Let f be any element in the closure of ΔNθWC(E). Then, there exists a sequence (fn) of points in ΔNθWC(E) such that limk→∞fk=f. To show that f is Nθ-ward continuous, take any Nθ-quasi-Cauchy sequence (αk) of points in E. Let ɛ>0. Since (fn) converges to f, there exists an n1∈N such that |f(αk)-fn1(αk)|<ɛ/3 for all k∈N. Hence,
(10)1hr∑k∈Ir|f(αk)-fn1(αk)|<1hr(kr-kr-1)ɛ3=ɛ3,
for r≥n1. As fn1 is Nθ-ward continuous on E, there exists a positive integer n2∈N such that r≥n2 implies that
(11)1hr∑k∈Ir|fn1(αk+1)-fn1(αk)|<ɛ3.
Now write n0=max{n1,n2}. Thus for r≥n0, we have
(12)1hr∑k∈Ir|f(αk+1)-f(αk)|≤1hr∑k∈Ir|f(αk+1)-fn1(αk+1)|+1hr∑k∈Ir|fn1(αk+1)-fn1(αk)|+1hr∑k∈Ir|fn1(αk)-f(αk)|<ɛ3+ɛ3+ɛ3=ɛ;
Hence,
(13)limr→∞1hr∑k∈Ir|f(αk+1)-f(αk)|=0.

Thus, f preserves Nθ-quasi-Cauchy sequences. This completes the proof of the theorem.

Corollary 8.

The set of all Nθ-ward continuous functions on a subset E of R is a complete subspace of the space of all continuous functions on E.

Proof.

The proof follows from the preceding theorem.

4. Conclusion

In this paper, new results concerning Nθ-ward continuity are obtained namely; a real function f defined on an interval E is uniformly continuous if and only if (f(αk)) is Nθ-quasi-Cauchy whenever (αk) is a quasi-Cauchy sequence of points in E, the uniform limit of Nθ-ward continuous functions is Nθ-ward continuous, and the set of all Nθ-ward continuous functions is a closed subset of the set of all continuous functions. We also prove that if a function f is uniformly continuous on a subset E of R, then, (f(αk)) is Nθ-quasi-Cauchy whenever (αk) is a quasi-Cauchy sequence of points in E.

As a further study one can find out if Theorem 3 is valid when the set E is replaced by a G-sequentially connected subset of R for a regular sequential method G [18]. For another further study, we suggest to investigate the present work for the fuzzy case. However, due to the change in settings, the definitions and methods of proofs will not always be analogous to those of the present work (see [19] for the definitions in the fuzzy setting). One can introduce and give an investigation of Nθ-quasi-Cauchy sequences in cone normed spaces (see [20] for basic concepts in cone normed spaces).

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