#### Abstract

We investigate the concept of Abel continuity. A function defined on a subset of , the set of real numbers, is Abel continuous if it preserves Abel convergent sequences. Some other types of continuities are also studied and interesting result is obtained. It turned out that uniform limit of a sequence of Abel continuous functions is Abel continuous and the set of Abel continuous functions is a closed subset of continuous functions.

#### 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 sciences, information theory, and dynamical systems.

A method of sequential convergence is a linear function defined on a linear subspace of , denoted by , into where and denote the set of real numbers and the space of all sequences, respectively. A sequence is said to be -convergent to if , and [1]. 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 . A function is called -continuous (see also [2, 3]) if for any -convergent sequence . Any matrix summability method on a subspace of is a method of sequential convergence. Abel summability method is a regular method of sequential convergence in this manner.

The purpose of this paper is to investigate the concept of Abel continuity for real functions and present interesting results.

#### 2. Definitions and Notations and Preliminary Results

We will use boldface letters ,,, for sequences , , of points in . A sequence of points in is called statistically convergent [4] (see also [5โ9]) to an element of if for every , and this is denoted by st-.

A sequence of points in is called lacunary statistically convergent [10] to an element of if for every , where , , as , and is an increasing sequence of positive integers, and this is denoted by - (see also [11โ13]). Throughout this paper we assume that . A sequence of points in is slowly oscillating [14] (see also [15, 16]), denoted by , if where denotes the integer part of .

A sequence of real numbers is called Abel convergent (or Abel summable) to if the series is convergent for and In this case we write Abel-. The set of Abel convergent sequences will be denoted by . Abel proved that if , then -; Abel sequential method is regular; that is, every convergent sequence is Abel convergent to the same limit ([17]; see also [18, 19]). As it is known that the converse is not always true in general, we see that the sequence is Abel convergent to but convergent in the ordinary sense.

*Definition 1. *A subset of is called Abel sequentially compact if, whenever is a sequence of points in , there is an Abel convergent subsequence of with .

*Definition 2. *A real number is said to be in the Abel sequentially closure of a subset of , denoted by , if there is a sequence of points in such that -, and it is called Abel sequentially closed if .

Note that the preceding definitions are special cases of the definitions of -sequential compactness and -sequential closure in [2].

It is clear that and . It is easily seen that . It is not always true that ; for example, . We note that any Abel sequentially closed subset of Abel sequentially compact subset of is also Abel sequentially compact. Thus intersection of two Abel sequentially compacts, Abel sequentially closed subsets of , is Abel sequentially compact. In general, any intersection of Abel sequentially compact and Abel sequentially closed subsets of is Abel sequentially compact. The condition closedness is essential here; that is, a subset of an Abel sequentially compact subset need not to be Abel sequentially compact. For example, the interval , that is, the set of real numbers strictly greater than and less than or equal to , is a subset of Abel sequentially compact subset , that is, the set of real numbers greater than or equal to and less than or equal to , but not Abel sequentially compact. Notice that union of two Abel sequentially compact subsets of is Abel sequentially compact. We see that any finite union of Abel sequentially compact subsets of is Abel sequentially compact, but any union of Abel sequentially compact subsets of is not always Abel sequentially compact.

#### 3. Main Results

First we introduce two notions.

*Definition 3. *A sequence of point in is called Abel quasi-Cauchy if is Abel convergent to ; that is,
where for each positive integer .

*Definition 4. *A subset of is called Abel ward compact if, whenever is a sequence of point in there is an Abel quasi-Cauchy subsequence of ; that is, .

We note that any Abel sequentially compact subset of is also Abel ward compact. Intersection of two Abel ward compact subsets of is Abel ward compact. In general, any intersection of Abel ward compact subsets of is Abel ward compact. Notice that union of two Abel ward compact subsets of is Abel ward compact. We see that any finite union of Abel sequentially compact subsets of is Abel ward compact, but any union of Abel ward compact subsets of is not always Abel ward compact.

Theorem 5. *A subset of is bounded if and only if it is Abel ward compact.*

*Proof. *It is an easy exercise to check that bounded subsets of are Abel ward compact. To prove that Abel ward compactness implies boundedness, suppose that is unbounded. If it is unbounded above, then one can construct a sequence of numbers in such that for each positive integer . Then the sequence does not have any Abel quasi-Cauchy subsequence, so is not Abel ward compact. If is bounded above and unbounded below, then similarly we obtain that is not Abel ward compact. This completes the proof.

We now introduce a new type of continuity defined via Abel convergent sequences.

*Definition 6. *A function is called Abel continuous, denoted by if it transforms Abel convergent sequences to Abel convergent sequences; that is, is Abel convergent to whenever is Abel convergent to .

We note that the sum of two Abel continuous functions is Abel continuous, and composite of two Abel continuous functions is Abel continuous, but the product of two Abel continuous functions need not be Abel continuous as it can be seen by considering product of the Abel continuous function with itself. We see that defined by - is a sequential method in the manner of [2], but not subsequential, so the theorems involving subsequentiality in [2] cannot be applied to Abel sequential method. In connection with Abel convergent sequences and convergent sequences the problem arises to investigate the following types of continuity of functions on :

We see that is Abel continuity of , and states the ordinary continuity of . We easily see that implies , implies , and implies . The converses are not always true as the identity function could be taken as a counter example for all the cases.

We note that can be replaced by either statistical continuity; that is, st- whenever is a statistically convergent sequence with st-, or lacunary statistical continuity; that is, - whenever is a lacunary statistically convergent sequence with -. More generally can be replaced by -sequential continuity of for any regular subsequential method .

Now we give the implication that implies ; that is, any Abel continuous function is continuous in the ordinary sense.

Theorem 7. *If a function is Abel continuous on a subset of , then it is continuous on in the ordinary sense.*

*Proof. *Suppose that a function is not continuous on . Then there exists a sequence with such that is not convergent to . If exists and is different from , then we easily see a contradiction. Now suppose that has two subsequences of such that and . Since is subsequence of , by hypothesis, and is a subsequence of , by hypothesis . This is a contradiction. If is unbounded above, then we can find an such that . There exists a positive integer an such that . Suppose that we have chosen an such that . Then we can choose an such that . Inductively we can construct a subsequence of such that . Since the sequence is a subsequence of , the subsequence is convergent and so is Abel convergent. But is not Abel convergent as we see in the line below. For each positive integer we have . The series is divergent for any satisfying and so is the series . This is a contradiction to the Abel convergence of the sequence . If is unbounded below, similarly is found to be divergent. The contradiction for all possible cases to the Abel continuity of completes the proof of the theorem.

The converse is not always true for the bounded function defined on as an example. The function is another example which is unbounded on as well.

On the other hand not all uniformly continuous functions are Abel continuous. For example, the function defined by is uniformly continuous but Abel continuous.

Corollary 8. *If f is Abel continuous, then it is statistically continuous.*

*Proof. *The proof follows from Theorem 7, Corollary 4 in [6], Lemma 1, and Theorem 8 in [3].

Corollary 9. *If f is Abel continuous, then it is lacunarily statistically sequentially continuous.*

*Proof. *The proof follows from Theorem 7 (see [20]).

Now we have the following result.

Corollary 10. *If is slowly oscillating, Abel convergent, and is an Abel continuous function, then is a convergent sequence.*

*Proof. *If is slowly oscillating and Abel convergent, then is convergent by the generalized Littlewood Tauberian theorem for the Abel summability method. By Theorem 7, is continuous, so is convergent, This completes the proof.

Corollary 11. *For any regular subsequential method , any Abel continuous function is -continuous.*

*Proof. *Let be an Abel continuous function and be a regular subsequential method. By Theorem 7, the function is continuous. Combining Lemma 1 and Corollary 9 of [3] we obtain that is -continuous.

For bounded functions we have the following result.

Theorem 12. *Any bounded Abel continuous function is Cesaro continuous.*

*Proof. *Let be a bounded Abel continuous function. Now we are going to obtain that is Cesaro continuous. To do this take any Cesaro convergent sequence with Cesaro limit . Since any Cesaro convergent sequence is Abel convergent to the same value [21] (see also [22]), is also Abel convergent to . By the assumption that is Abel continuous, is Abel convergent to . By the boundedness of , is bounded. By Corollary to Karamata's Hauptsatz on page 108 in [18], is Cesaro convergent to . Thus is Cesaro continuous at the point . Hence is Cesaro continuous at any point in the domain.

Corollary 13. *Any bounded Abel continuous function is uniformly continuous.*

*Proof. *The proof follows from the preceding theorem, and the theorem on page 73 in [23].

It is well known that uniform limit of a sequence of continuous functions is continuous. This is also true for Abel continuity; that is, uniform limit of a sequence of Abel continuous functions is Abel continuous.

Theorem 14. *If is a sequence of Abel continuous functions defined on a subset E of and is uniformly convergent to a function , then is Abel continuous on .*

*Proof. *Let be an Abel convergent sequence of real numbers in . Write . Take any . Since is uniformly convergent to , there exists a positive integer such that
for all whenever . Hence
As is Abel continuous, then there exist a for such that
Now for , it follows from (6), (7), and (8) that

This completes the proof of the theorem.

In the following theorem we prove that the set of Abel continuous functions is a closed subset of the space of continuous functions.

Theorem 15. *The set of Abel 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 Abel 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 Abel continuous, take any Abel convergent sequence of points with Abel limit . Let . Since is convergent to , there exists a positive integer such that
for all whenever . Write
and . Then we obtain that for any satisfying ,
As is Abel continuous, then there exists a such that for
Let . Now, for , we have

This completes the proof of the theorem.

Corollary 16. *The set of all Abel 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 and the fact that the set of all continuous functions on is complete.

Theorem 17. *Abel continuous image of any Abel sequentially compact subset of is Abel sequentially compact.*

*Proof. *Although the proof follows from Theorem 7 in [2], we give a short proof for completeness. Let be any Abel continuous function defined on a subset of and let be any Abel sequentially compact subset of . Take any sequence of point in . Write for each positive integer . Since is Abel sequentially compact, there exists an Abel convergent subsequence of the sequence . Write -. Since is Abel continuous -. Thus is Abel convergent to and a subsequence of the sequence . This completes the proof.

For -, we have the following.

Theorem 18. *If a function is Abel continuous on a subset of , then
**
for every subset of .*

*Proof. *The proof follows from the regularity of Abel method and Theorem 8 on page 316 of [3].

Theorem 19. *For any regular subsequential method , if a subset of is -sequentially compact, then it is Abel sequentially compact.*

*Proof. *The proof can be obtained by noticing the regularity and subsequentiality of (see [2] for the detail of -sequential compactness).

Theorem 20. *A subset ofโโ is Abel sequentially compact if and only if it is bounded and closed.*

*Proof. *It is clear that any bounded and closed subset of is Abel sequentially compact. Suppose first that is unbounded so that we can construct a sequence of points in such that and for each positive integer . It is easily seen that the sequence has no Abel convergent subsequence. If it is unbounded below, then similarly we construct a sequence of points in which has no Abel convergent subsequence. Hence is not Abel sequentially compact. Now suppose that is not closed so that there exists a point in . Then there exists a sequence of of points in that converges to . Every subsequence of also converges to . Since Abel method is regular, every subsequence of Abel converges to . Since is not a member of , is not Abel sequentially compact. This contradiction completes the proof that Abel sequentially compactness implies boundedness and closedness.

We note that an Abel sequentially compact subset of is slowly oscillating compact [15], an Abel sequentially compact subset of is ward compact [24], and an Abel sequentially compact subset of is -ward compact [25].

#### 4. Conclusion

In this paper we introduce a concept of Abel continuity and a concept of Abel sequential compactness and present theorems related to this kind of sequential continuity, this kind of sequential compactness, continuity, statistical continuity, lacunary statistical continuity, and uniform continuity. One may expect this investigation to be a useful tool in the field of analysis in modeling various problems occurring in many areas of science, dynamical systems, computer science, information theory, and biological science. On the other hand, we suggest to introduce a concept of fuzzy Abel sequential compactness and investigate fuzzy Abel continuity for fuzzy functions (see [26] for the definitions and related concepts in fuzzy setting). However due to the change in settings, the definitions and methods of proofs will not always be the same. We also suggest to investigate a theory in dynamical systems by introducing the following concept: two dynamical systems are called Abel-conjugate if there is a one-to-one and onto function such that, , and are Abel continuous, and commutes the mappings at each point. An investigation of Abel continuity and Abel compactness can be done for double sequences (see [27] for basic concepts in the double sequences case). It seems both double Abel continuity and Abel continuity coincides but it needs proving.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

The authors would like to thank the referees for a careful reading and several constructive comments that have improved the presentation of the results.