A Study on N θ-Quasi-Cauchy Sequences

and Applied Analysis 3 fix z 0 , z k 1 , . . . , z k n k inEwith z 0 = ηk, z k n k = ξk+1, and |z k i −z k i−1 | < 1/k for 1 ≤ i ≤ nk. Now write (ξ1, η1, z 1 1 , . . . , z 1 n 1 −1 , ξ2, η2, z 2 1 , . . . , z 2 n 2 −1 , ξ3, η3, . . . , ξk, ηk, z k 1 , . . . , z k n k−1 , ξk+1, ηk+1, . . .) . (5) 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 quasiCauchy 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 Nθward continuous, then, it is uniformly continuous. Proof. Suppose that f is not uniformly continuous on E. Then, there is an ε0 > 0 such that for any δ > 0 there exist x, y ∈ E with |x − y| < δ but |f(x) − f(y)| ≥ ε0. For every integer n ≥ 1 fix ξn, ηn ∈ E with |ξn − ηn| < 1/n and |f(ξn) − f(ηn)| ≥ ε0. By the lemma, there exists an Nθquasi-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)| ≥ ε0; hence, (f(αi)) is notNθ-quasi-Cauchy. Thus, f does not preserveNθ-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 withTheorem 1, we have that a real function f defined on an interval E is uniformly continuous if and only if (f(αk)) isNθ-quasi-Cauchy whenever (αk) is a quasiCauchy 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 fromTheorem 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 fromTheorem 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 ofNθ-ward continuity; that is, uniform limit of a sequence of Nθ-ward continuous functions isNθ-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 isNθ-ward continuous on E. Proof. Let (αk) be any Nθ-quasi-Cauchy sequence of points in E, and let ε be any positive real number. By uniform convergence of (fn), there exists an n1 ∈ N such that |f(α) − fk(α)| < ε/3 for n ≥ n1 and every α ∈ E. Hence,


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  is continuous if and only if it preserves convergent sequences.A subset  of R, the set of real numbers, is compact if any sequence of points in  has a convergent subsequence whose limit is in .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], quasislowly oscillating continuity, quasi-slowly oscillating compactness [2], Δ-quasi-slowly oscillating continuity, Δ-quasislowly oscillating compactness [3][4][5], ward continuity, ward compactness [6,7], -ward continuity, -ward compactness [8], statistical ward continuity, and lacunary statistical ward continuity [9,10].
In [11], the notion of   convergence was introduced, and studied by Freedman et al.Using the main idea for continuity and compactness given above the concepts of  ward compactness of a subset  of R and   -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  ward continuity.

Preliminaries
Boldface letters , x, y, z, . . .will be used for sequences  = (  ),  = (  ), y = (  ), z = (  ), . . . of points in the set of real numbers R for the sake of abbreviation.Sums of the form ∑    −1 +1 |  | frequently occur and will often be written for convenience as 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 (  ) of points in R is quasi-Cauchy if (Δ  ) is a null sequence where Δ  =  +1 −   .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].
where   = ( −1 ,   ] and  = (  ) is a lacunary sequence, that is, an increasing sequence of positive integers such that  0 = 0 and ℎ  :   − where [] denotes the integer part of  (see also [14]).An ideal  is a family of subsets of positive integers N which is closed under taking finite unions and subsets of its elements.A sequence (  ) of real numbers is said to be ideal convergent to a real number , if for each  > 0 the set { : |  − | ≥ } belongs to .Ideal ward compactness of a subset of R and ideal ward continuity of a real function were recently introduced by C ¸akalli and Hazarika in [15].

Results
Any quasi-Cauchy sequence is   -quasi-Cauchy, so any slowly oscillating sequence is   -quasi-Cauchy, and so any Cauchy sequence is.A sequence  = (  ) is called Cesaro summable to a real number  if lim  → ∞ (1/) ∑  =1   = .This is denoted by  1 − lim   = , and the set of all Cesaro sequences is denoted by 1 | ⊂ Δ 0 1 .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  quasi-Cauchy sequences (see also [16]).Combining these facts, for any lacunary sequence , we have the following: In the sequel, we will always assume that lim inf    > 1.
We observe that   -summability is a kind of strong summability where  = (  ) is a regular matrix generated by the lacunary sequence  = (  ) as follows: On the other hand, we see that   -ward continuity cannot be given as a strong -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   -ward continuous function is ideal continuous; furthermore any   continuous function is ideal continuous for an admissible ideal.
Theorem 1.If a function  is uniformly continuous on a subset  of R, then, ((  )) is   -quasi-Cauchy whenever (  ) is a quasi-Cauchy sequence of points in .
Proof.Let  be a subset of R, and let  be a uniformly continuous function on .Take any quasi-Cauchy sequence (  ) of points in , and let  be any positive real number.By uniform continuity of , there exists a  > 0 such that |() − ()| <  whenever | − | <  and ,  ∈ .Since (  ) is a quasi-Cauchy sequence, there exists a positive integer  0 such that | +1 −   | <  for  ≥  0 .Hence, for  ≥  0 .Thus, ((  )) is an   -quasi-Cauchy sequence.This completes the proof of the theorem.
We have much more below for a real function  defined on an interval that  is uniformly continuous if and only if ((  )) is   -quasi-Cauchy whenever (  ) is a quasi-Cauchy sequence of points in .First, we give the following lemma.Proof.Although the following proof is similar to that of [13] Observing that the sequence, constructed in the proof of the preceding theorem, is also a quasi-Cauchy sequence, we obtain that a real function  defined on an interval  is uniformly continuous if ((  )) is   -quasi-Cauchy whenever (  ) is a quasi-Cauchy sequence of points in .Combining this with Theorem 1, we have 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 .

Corollary 4. If a function defined on an interval is 𝑁 𝜃 -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 𝑁 𝜃 -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   -ward continuity; that is, uniform limit of a sequence of   -ward continuous functions is   -ward continuous.Theorem 6.If (  ) is a sequence of   -ward continuous functions on a subset  of R and (  ) is uniformly convergent to a function , then,  is   -ward continuous on .
Proof.Let (  ) be any   -quasi-Cauchy sequence of points in , and let  be any positive real number.By uniform convergence of (  ), there exists an  1 ∈ N such that |() −   ()| < /3 for  ≥  1 and every  ∈ .Hence, for  ≥  1 and every  ∈ .As   1 is   -ward continuous on , there exists an  2 ∈ N such that for  ≥  2 Now write  0 = max{ 1 ,  2 }.Thus for  ≥  0 , we have Thus,  preserves   -quasi-Cauchy sequences.This completes the proof of the theorem.

Corollary 8.
The set of all   -ward continuous functions on a subset  of R is a complete subspace of the space of all continuous functions on .
Proof.The proof follows from the preceding theorem.

Conclusion
In this paper, new results concerning   -ward continuity are obtained namely; 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 , 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 prove that if a function  is uniformly continuous on a subset  of R, then, ((  )) is   -quasi-Cauchy whenever (  ) is a quasi-Cauchy sequence of points in .
As a further study one can find out if Theorem 3 is valid when the set  is replaced by a -sequentially connected subset of R for a regular sequential method  [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   -quasi-Cauchy sequences in cone normed spaces (see [20] for basic concepts in cone normed spaces).

Lemma 2 .
If (  ,   ) is a sequence of ordered pairs of points in an interval such that lim  → ∞ |  −   | = 0, then, there exists an   -quasi-Cauchy sequence (  ) with the property that for any positive integer  there exists a positive integer  such that (  ,   ) = ( −1 ,   ).
−1 → ∞.The intervals determined by  are denoted by   = ( −1 ,   ], and the ratio   / −1 is abbreviated by  .A sequence (  ) of points in R is called   -quasi-Cauchy if (Δ  ) is   -convergent to 0. A function defined on a subset  of R is called   -ward continuous if it preserves   -quasi-Cauchy sequences, that is, ((  )) is an   -quasi-Cauchy sequence whenever (  ) is. |Δ 0  | will denote the set of   -quasi-Cauchy sequences of points in R. Any subsequence of a Cauchy sequence is Cauchy.The analogous property fails for   -quasi-Cauchy sequences.A counterexample for the case is the sequence (  ) = (√) with the subsequence (  2 ) = ().
and only if lim sup   < ∞ for any lacunary sequence .
The set of all   -ward continuous functions on a subset  of R is a closed subset of the set of all continuous functions on , that is, Δ  () = Δ  (), where Δ  () is the set of all   -ward continuous functions on  and Δ  () denotes the set of all cluster points of Δ  ().Proof.Let  be any element in the closure of Δ  ().Then, there exists a sequence (  ) of points in Δ  () such that lim  → ∞   = .To show that  is   -ward continuous, take any   -quasi-Cauchy sequence (  ) of points in .Let  > 0. Since (  ) converges to , there exists an  1 ∈ N such that |(  ) −   1 (  )| < /3 for all  ∈ N. Now write  0 = max{ 1 ,  2 }.Thus for  ≥  0 , we have for  ≥  1 .As   1 is   -ward continuous on , there exists a positive integer  2 ∈ N such that  ≥  2 implies that1 ℎ  ∑ ∈         1 ( +1 ) −   1 (  )