#### Abstract

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 .

#### 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 is continuous if and only if it preserves convergent sequences. A subset of , 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], quasi-slowly oscillating continuity, quasi-slowly oscillating compactness [2], -quasi-slowly oscillating continuity, -quasi-slowly oscillating compactness [3–5], 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 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.

#### 2. Preliminaries

Boldface letters , , , will be used for sequences , , , of points in the set of real numbers for the sake of abbreviation. Sums of the form 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 is quasi-Cauchy if is a null sequence where . These sequences were named as quasi-Cauchy by Burton and Coleman [13, page 328], while they were called as forward convergent to sequences in [7, page 226]. A sequence of points in is called -convergent to an element of if where and is a lacunary sequence, that is, an increasing sequence of positive integers such that and . The intervals determined by are denoted by , and the ratio is abbreviated by . A sequence of points in is called -quasi-Cauchy if is -convergent to . A function defined on a subset of is called -ward continuous if it preserves -quasi-Cauchy sequences, that is, is an -quasi-Cauchy sequence whenever is. will denote the set of -quasi-Cauchy sequences of points in . 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 .

A sequence of points in is slowly oscillating [14, Definition 2, page 947] if where denotes the integer part of (see also [14]).

An ideal is a family of subsets of positive integers 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 the set belongs to . Ideal ward compactness of a subset of and ideal ward continuity of a real function were recently introduced by Çakalli and Hazarika in [15].

#### 3. 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 . This is denoted by , and the set of all Cesaro sequences is denoted by . We call a sequence Cesaro quasi-Cauchy if . The set of all Cesaro quasi-Cauchy sequences is denoted by . A sequence is called strongly Cesaro summable to a real number if . This is denoted by . The set of all strongly Cesaro summable sequences is denoted by . We call a sequence strongly Cesaro quasi-Cauchy if . The set of all strongly Cesaro quasi-Cauchy sequences is denoted by . The following inclusions are satisfied: and .

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]).(i) if and only if for any lacunary sequence .(ii) if and only if for any lacunary sequence .

Combining these facts, for any lacunary sequence , we have the following:(iii) if and only if ;(iv) if and only if .

In the sequel, we will always assume that .

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 , then, is -quasi-Cauchy whenever is a quasi-Cauchy sequence of points in .*

* Proof. *Let be a subset of , 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 such that whenever and . Since is a quasi-Cauchy sequence, there exists a positive integer such that for . Hence,
for . 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.

Lemma 2. *If is a sequence of ordered pairs of points in an interval such that , then, there exists an -quasi-Cauchy sequence with the property that for any positive integer there exists a positive integer such that .*

* Proof. *Although the following proof is similar to that of [13], we give it for completeness. For each positive integer , we can fix in with , , and for . Now write
Then denoting this sequence by , we obtain that for any positive integer there exists a positive integer such that . The sequence constructed is a quasi-Cauchy sequence, and it is an -quasi-Cauchy sequence, since any quasi-Cauchy sequence is an -quasi-Cauchy sequence. This completes the proof of the lemma.

Theorem 3. *If a function defined on an interval is -ward continuous, then, it is uniformly continuous.*

* Proof. *Suppose that is not uniformly continuous on . Then, there is an such that for any there exist with but . For every integer fix , with and . By the lemma, there exists an -quasi-Cauchy sequence such that for any integer there exists a with and . This implies that ; hence, is not -quasi-Cauchy. Thus, does not preserve -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 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 -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 -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 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 such that for and every . Hence,
for and every . As is -ward continuous on , there exists an such that for
Now write . Thus for , we have
Hence,
Thus, preserves -quasi-Cauchy sequences. This completes the proof of the theorem.

Theorem 7. *The set of all -ward continuous functions on a subset of is a closed subset of the set of all continuous functions on , that is, , where is the set of all -ward continuous functions on and denotes the set of all cluster points of .*

* Proof. *Let be any element in the closure of . Then, there exists a sequence of points in such that . To show that is -ward continuous, take any -quasi-Cauchy sequence of points in . Let . Since converges to , there exists an such that for all . Hence,
for . As is -ward continuous on , there exists a positive integer such that implies that
Now write . Thus for , we have
Hence,

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 is a complete subspace of the space of all continuous functions on .*

* Proof. *The proof follows from the preceding theorem.

#### 4. 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 , 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 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).