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Abstract and Applied Analysis
Volume 2010, Article ID 258980, 6 pages
http://dx.doi.org/10.1155/2010/258980
Research Article

New Types of Continuities

1Department of Mathematics, Ege University, Bornova, İzmir 35100, Turkey
2Rockford College, 5050 E. State Street, Rockford, IL 61108, USA

Received 18 February 2010; Accepted 5 April 2010

Academic Editor: Ferhan Atici

Copyright © 2010 İbrahim Çanak and Mehmet Dik. 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 new concept of quasi slowly oscillating continuity is introduced. Furthermore, it is shown that this kind of continuity implies ordinary continuity, but the converse is not always true.

1. Introduction

Firstly, some definitions and notations will be given in the following. Throughout this paper, will denote the set of all positive integers. We will use boldface letters , , for sequences , , and of terms in , the set of all real numbers. Also, and will denote the set of all sequences of points in and the set of all convergent sequences of points in , respectively.

A sequence of points in is called statistically convergent [1] to an element of if for every , and this is denoted by .

A sequence of points in is slowly oscillating [2], denoted by , if where denotes the integer part of . This is equivalent to the following: whenever as . In terms of and , this is also equivalent to the case when for any given , there exist and the positive integer such that if and .

By a method of sequential convergence, or briefly a method, we mean a linear function defined on a sublinear space of , denoted by , into . A sequence is said to be -convergent [3] 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 subsequentially convergent if whenever is -convergent with , then there is a subsequence of with . A function is called -continuous [3] if for any -convergent sequence . Here we note that for special , is called statistically continuous [3].

Our goal in this paper is to introduce a new concept of quasi slowly oscillating continuity, which cannot be given by means of any as in [3]. It is proved that quasi slowly oscillating continuity implies ordinary and statistical continuities. We also introduce several other types of continuities and a new type of compactness.

2. Slowly Oscillating Continuity

In [4], the concept of slowly oscillating continuity is defined in the sense that a function is slowly oscillating continuous, denoted by , if it transforms slowly oscillating sequences to slowly oscillating sequences, that is, is slowly oscillating whenever is slowly oscillating.

We note that slowly oscillating continuity cannot be obtained by any sequential method .

In the following theorem, we prove that the set of slowly oscillating continuous functions is a closed subset of the space of all continuous functions.

Theorem 2.1. The set of slowly oscillating 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 slowly oscillating 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 slowly oscillating continuous, take any slowly oscillating sequence . Let . Since converges to , there exists a positive integer such that for all and for all , . As is slowly oscillating continuous, there exist a positive integer and a such that if and . Hence, for all and , This completes the proof of the theorem.

Corollary 2.2. The set of all slowly oscillating continuous functions on a subset of is a complete subspace of the space of all continuous functions on .

Proof. The proof follows from Theorem 2.1.

The following theorem shows that on a slowly oscillating compact subset of R, slowly oscillating continuity implies uniform continuity.

Theorem 2.3. Let be a slowly oscillating compact subset of and let be slowly oscillating continuous on . Then is uniformly continuous on .

Proof. Assume that is not uniformly continuous on . Then there exist and sequences and in such that for all . Since is slowly oscillating compact, there is a slowly oscillating subsequence of . It is clear that the corresponding sequence is also slowly oscillating, since If is slowly oscillating continuous, then the sequences and are slowly oscillating. By (2.2), this is not possible. Consequently, we obtain that if is slowly oscillating continuous on a slowly oscillating compact set of , then is uniformly continuous on .

Corollary 2.4. For any regular subsequential method , any slowly oscillating continuous function is -continuous.

3. Quasi slowly Oscillating Continuity

A sequence is called quasi slowly oscillating, denoted by , if is a slowly oscillating sequence. The concept of slowly oscillating continuity [4] suggests a new kind of continuity. A function is quasi slowly oscillating continuous if it transforms quasi slowly oscillating sequences to quasi slowly oscillating sequences, that is, is quasi slowly oscillating whenever is quasi slowly oscillating.

Notice that any slowly oscillating sequence is quasi slowly oscillating, but the converse is not always true. Indeed, let be slowly oscillating. Hence, from the definition of slow oscillation of a sequence and the line below we immediately have that is quasi slowly oscillating. To see that the converse is not true, we consider the following example. The sequence defined by is quasi slowly oscillating, but not slowly oscillating. We see that composition of two quasi slowly oscillating continuous functions is quasi slowly oscillating continuous. It is clear that the sum of two quasi slowly oscillating continuous functions is quasi slowly oscillating continuous. The product of two quasi slowly oscillating continuous functions needs not be quasi slowly oscillating. For example, for the quasi slowly oscillating continuous functions and defined by and , the function is not quasi slowly oscillating. When both and are bounded and quasi slowly oscillating continuous, we have the following theorem.

Theorem 3.1. If and are bounded quasi slowly oscillating continuous functions on a subset of , then their product is quasi slowly oscillating continuous on .

This follows from the definition of quasi slowly oscillating continuity.

In connection with slowly oscillating sequences, quasi slowly oscillating sequences, and convergent sequences, the problem arises to investigate the following types of continuity of functions on . . .(c-c) For each in the domain of , whenever . .. . is uniform continuity of .

It is clear that (QSO-QSO) implies (SO-QSO), but (SO-QSO) needs not imply (QSO-QSO). Also (QSO-c) implies (c-QSO) and (QSO-c) implies (c-c) and we see that (c-c) needs not imply (QSO-c) since the identity function is an example. We also note that (u) implies (SO-QSO).

Theorem 3.2. If is quasi slowly oscillating continuous on , then it is continuous in the ordinary sense.

Proof. Let be any convergent sequence with . Then the sequence also converges to and is quasi slowly oscillating. By the hypothesis, the sequence is slowly oscillating. It follows from the definition of slow oscillation that and . Hence, we have . This completes the proof.

Corollary 3.3. Any quasi slowly oscillating continuous function is G-continuous for any regular subsequential method G.

Corollary 3.4. Any quasi slowly oscillating continuous function is statistically continuous.

It is well known that uniform limit of a sequence of continuous functions is continuous. This is also true for quasi slowly oscillating continuity, that is, uniform limit of a sequence of quasi slowly oscillating continuous functions is quasi slowly oscillating continuous.

Theorem 3.5. If is a sequence of quasi slowly oscillating continuous functions defined on a subset of and let is uniformly convergent to a function , then is quasi slowly oscillating continuous on .

Proof. Let be a sequence of quasi slowly oscillating continuous functions defined on a subset of and be uniformly convergent to a function . Let be a quasi slowly oscillating sequence and . As is uniformly convergent to , there exists a positive integer such that for all whenever . Since is quasi 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.

Definition 3.6. A subset of is called slowly oscillating compact [4] if whenever is a sequence of points in , there is a slowly oscillating subsequence of .

Firstly, we see that quasi slowly oscillating compactness cannot be obtained by any -sequential compactness in the manner of [5].

We note that any compact subset of is quasi slowly oscillating compact and any subset of a quasi slowly oscillating compact subset of is also quasi slowly oscillating compact. Thus, intersection of two quasi slowly oscillating compact subsets of is quasi slowly oscillating compact. More generally any intersection of quasi slowly oscillating compact subsets of is quasi slowly oscillating compact.

We also note that for any regular subsequential method , any -sequentially compact subset of is quasi slowly oscillating compact.

Notice that the union of two quasi slowly oscillating compact subsets of is quasi slowly oscillating compact. We see that any finite union of quasi slowly oscillating compact subsets of is quasi slowly oscillating compact, but any union of quasi slowly oscillating compact subsets of is not always quasi slowly oscillating compact. For example, consider the sets for . Then for each constant , the set is quasi slowly oscillating compact, but is not quasi slowly oscillating compact.

Theorem 3.7. Let be a quasi slowly oscillating compact subset of and let be a quasi slowly oscillating continuous function. Then is quasi slowly oscillating compact.

Proof. Let be any sequence of points in . Then there exists a sequence such that for each . As is quasi slowly oscillating compact, there exists a quasi slowly oscillating subsequence of the sequence . Since is quasi slowly oscillating continuous, is quasi slowly oscillating. It follows that is quasi slowly oscillating compact.

Corollary 3.8. Quasi slowly oscillating continuous image of any compact subset of is compact.

Theorem 3.9. The set of quasi slowly oscillating 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 slowly oscillating 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 quasi slowly oscillating continuous, take any quasi slowly oscillating sequence . Let . Since converges to , there exists a positive integer such that for all and for all , . As is quasi slowly oscillating continuous, there exist a positive integer and a such that if and . Hence, for all , This completes the proof of the theorem.

Finally we note that the following further investigation problems arose.

Problem 1. For further study, we suggest to investigate quasi slowly oscillating sequences of fuzzy points and quasi slowly oscillating continuity for the fuzzy functions. 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, e.g., [6]).

Problem 2. Investigate a theory in dynamical systems by introducing the following concept: two dynamical systems are called pseudo conjugate if there is 1-1, onto, pseudo continuous , such that is -pseudo continuous, and commutes the mappings at each point .

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