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Abstract and Applied Analysis
VolumeΒ 2008, Article IDΒ 485706, 5 pages
http://dx.doi.org/10.1155/2008/485706
Research Article

Slowly Oscillating Continuity

Department of Mathematics, Faculty of Science and Letters, Maltepe University, 34857 Maltepe, Istanbul, Turkey

Received 2 November 2007; Accepted 11 February 2008

Academic Editor: FerhanΒ Atici

Copyright Β© 2008 H. Γ‡akalli. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A function is continuous if and only if, for each point in the domain, , whenever . This is equivalent to the statement that is a convergent sequence whenever is convergent. The concept of slowly oscillating continuity is defined in the sense that a function is slowly oscillating continuous if it transforms slowly oscillating sequences to slowly oscillating sequences, that is, is slowly oscillating whenever is slowly oscillating. A sequence of points in is slowly oscillating if , where denotes the integer part of . Using 's and 's, this is equivalent to the case when, for any given , there exist and such that if and . A new type compactness is also defined and some new results related to compactness are obtained.

1. Introduction

The notion of statistical convergence was introduced by Fast [1] and has been investigated in [2]. In [3], Zygmund called it β€œalmost convergence” and established a relation between it and strong summability.

Following the idea given in the 1946 American Mathematical Monthly problem [4], a number of authors including Posner [5], IwiΕ„ski [6], Srinivasan [7], Antoni [8], Antoni and Ε alΓ‘t [9], Spigel and Krupnik [10] have studied continuity defined by a regular summability matrix . Some authors (Γ–ztΓΌrk [11], Savaş and Das [12], BorsΓ­k and Ε alΓ‘t [13]) have studied continuity for methods of almost convergence or for related methods.

Recently, Connor and Grosse-Erdman [14] have given sequential definitions of continuity for real functions calling it continuity instead of continuity and their results cover the earlier works related to continuity where a method of sequential convergence, or briefly a method, is a linear function defined on a linear subspace of , denoted by , into where is the set of all sequences of points in . A sequence is said to be convergent to if and . In particular, denotes the limit function on the linear space . A method is called regular if every convergent sequence is convergent with . A method is called subsequential if whenever is convergent with then there is a subsequence of with .

The purpose of this paper is to introduce a concept of slowly oscillating continuity which cannot be given by means of any as in [14] and prove that slowly oscillating continuity implies ordinary continuity and so statistical continuity and is implied by uniform continuity; and introduce several other types of continuities; and introduce some new types of compactness.

2. Slowly Oscillating Continuity

It is known that a sequence of points in is slowly oscillating if in which denotes the integer part of . Using 's and 's, this is equivalent to the case when for any given , there exists and such that if and . It is well known that a function is continuous if and only if, for each point in the domain, whenever . This is equivalent to the statement that is a convergent sequence whenever is convergent. As a result of completeness, when the domain of the function is all of , or a complete subset of , this is also equivalent to the fact that is a Cauchy sequence whenever is Cauchy. This suggests that we might introduce a concept of slowly oscillating continuity as defined in the sense that a function is slowly oscillating continuous if it transforms slowly oscillating sequences to slowly oscillating sequences, that is, is slowly oscillating whenever is slowly oscillating.

We note that the sum of two slowly oscillating continuous functions is slowly oscillating continuous and that the composite of two slowly oscillating continuous functions is slowly oscillating continuous; but the product of two slowly oscillating continuous functions needs not be slowly oscillating continuous as can be seen by considering product of the slowly oscillating function with itself.

In connection with slowly oscillating sequences and convergent sequences, the problem arises to investigate the following types of continuity of functions on where denotes the set of all convergent sequences and denotes the set of all slowly oscillating sequences of points in .

(w)Slowly oscillating continuity of on .(wc).(c).(cp1) whenever .(d).(u)Uniform continuity of on .

It is clear that (c) is equivalent to (cp1) for each . It is easy to see that (wc) implies (d); (wc) implies (w); and (w) implies (d). Now we give a proof of the implication (w) implies (c) in the following.

Theorem 2.1. If is slowly oscillating continuous on a subset of , then it is continuous on in the ordinary sense.

Proof. Let be any convergent sequence with . Then the sequence also converges to . Then the sequence is slowly oscillating hence, by the hypothesis, the sequence is slowly oscillating. It follows from this that converges to , so the sequence also converges to . This completes the proof.

The converse is not always true for the function is an example.

Corollary 2.2. If is slowly oscillating continuous, then it is statistically continuous.

Corollary 2.3. If is statistically convergent and slowly oscillating and is a slowly oscillating continuous function, then is a convergent sequence.

Theorem 2.4. If a function on a subset of is uniformly continuous, then it is slowly oscillating continuous on .

Proof. Let be a uniformly continuous function and let be any slowly oscillating sequence of points in . To prove that is slowly oscillating, take any . Uniform continuity of implies that there exists a such that , whenever . Since is slowly oscillating, for this , there exist a and such that if and . Hence if and . It follows from this that is slowly oscillating.

It is well known that any continuous function on a compact set is also uniformly continuous. It is also true for a regular subsequential method that any continuous function on a sequentially compact set is also uniformly continuous (see [15] for the definition of compactness).

Theorem 2.5. If is a sequence of slowly oscillating continuous functions defined on a subset of and is uniformly convergent to a function , then is slowly oscillating continuous on .

Proof. Let be a slowly oscillating sequence and . Then there exists a positive integer such that for all , whenever . As is slowly oscillating continuous, there exist a and a positive integer such that for and . Now for and , we have This completes the proof of the theorem.

Now we can give the definition of slowly oscillating compactness of a subset of .

Definition 2.6. A subset of is called slowly oscillating compact if whenever is a sequence of points in there is a slowly oscillating subsequence of .
Any compact subset of is slowly oscillating compact. Union of two slowly oscillating compact subsets of is slowly oscillating compact. We note that any subset of a slowly oscillating compact set is also slowly oscillating compact, and so intersection of any slowly oscillating compact subsets of is slowly oscillating compact.

Theorem 2.7. For any regular subsequential method , if a subset of is sequentially compact, then it is slowly oscillating compact.

Proof. The proof can be obtained by noticing the regularity and subsequentiality of (see [15] for the detail of compactness).

The converse is not necessarily true, for example, is slowly oscillating compact, but it is not sequentially compact when on .

Theorem 2.8. Slowly oscillating continuous image of any slowly oscillating compact subset of is slowly oscillating compact.

Proof. Let be a slowly oscillating continuous function on and let be a slowly oscillating compact subset of . Take any sequence of points in . Write for each . As is assumed to be slowly oscillating compact, there exists a subsequence of the sequence for which is slowly oscillating. Since is slowly oscillating continuous, the image of the sequence , is slowly oscillating. Since is a subsequence of the sequence , the proof is completed.

We add one more compactness defining as saying that a subset of is called Cauchy compact if whenever is a sequence of points in , there is a subsequence of which is Cauchy; we see that any Cauchy compact subset of is also slowly oscillating compact, and a slowly oscillating continuous image of any Cauchy compact subset of is Cauchy compact.

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