New Types of Continuities

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, 𝐍 will denote the set of all positive integers. We will use boldface letters 𝐱, 𝐲, 𝐳,… for sequences 𝐱=(𝑥_𝑛), 𝐲=(𝑦_𝑛), and 𝐳=(𝑧_𝑛),… of terms in 𝐑, the set of all real numbers. Also, 𝐬 and 𝐜 will denote the set of all sequences of points in 𝐑 and the set of all convergent sequences of points in 𝐑, respectively.

A sequence 𝐱=(𝑥_𝑛) of points in 𝐑 is called statistically convergent [1] to an element ℓ of 𝐑 if lim_𝑛→∞1𝑛|{𝑘≤𝑛∶|𝑥_𝑘−ℓ|≥𝜀}|=0, (1.1) for every 𝜀>0, and this is denoted by 𝑠𝑡−lim_𝑛→∞𝑥_𝑛=ℓ.

A sequence 𝐱=(𝑥_𝑛) of points in 𝐑 is slowly oscillating [2], denoted by 𝐱∈𝐒𝐎, if lim_𝜆→1+lim_𝑛max_{𝑛+1≤𝑘≤[𝜆𝑛]}|𝑥_𝑘−𝑥_𝑛|=0, (1.2) where [𝜆𝑛] denotes the integer part of 𝜆𝑛. This is equivalent to the following: 𝑥_𝑚−𝑥_𝑛→0 whenever 1≤𝑚/𝑛→1 as 𝑚,𝑛→∞. In terms of 𝜀>0 and 𝛿, this is also equivalent to the case when for any given 𝜀>0, there exist 𝛿=𝛿(𝜀)>0 and the positive integer 𝑁=𝑁(𝜀) such that |𝑥_𝑚−𝑥_𝑛|<𝜀 if 𝑛≥𝑁(𝜀) and 𝑛≤𝑚≤(1+𝛿)𝑛.

By a method of sequential convergence, or briefly a method, we mean a linear function 𝐺 defined on a sublinear space of 𝐬, denoted by 𝑐_𝐺(𝐑), into 𝐑. A sequence 𝐱=(𝑥_𝑛) is said to be 𝐺-convergent [3] to ℓ if 𝐱∈𝑐_𝐺(𝐑) and 𝐺(𝐱)=ℓ. In particular, lim denotes the limit function lim𝐱=lim_𝑛𝑥_𝑛 on the linear space 𝐜. A method 𝐺 is called regular if every convergent sequence 𝐱=(𝑥_𝑛) is 𝐺-convergent with 𝐺(𝐱)=lim𝐱. A method 𝐺 is called subsequentially convergent if whenever 𝐱 is 𝐺-convergent with 𝐺(𝐱)=ℓ, then there is a subsequence (𝑥_𝑛𝑘) of 𝐱 with lim_𝑘𝑥_𝑛𝑘=ℓ. A function 𝑓 is called 𝐺-continuous [3] if 𝐺(𝑓(𝐱))=𝑓(𝐺(𝐱)) for any 𝐺-convergent sequence 𝐱. Here we note that for special 𝐺=𝑠𝑡−lim, 𝑓 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 𝐺 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 𝑓 is slowly oscillating continuous, denoted by 𝑓∈𝐒𝐎𝐂, if it transforms slowly oscillating sequences to slowly oscillating sequences, that is, (𝑓(𝑥_𝑛)) is slowly oscillating whenever (𝑥_𝑛) is slowly oscillating.

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

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 𝐸 of 𝐑 is a closed subset of the set of all continuous functions on 𝐸, that is, 𝐒𝐎𝐂(𝐸)=𝐒𝐎𝐂(𝐸), where 𝐒𝐎𝐂(𝐸) is the set of all slowly oscillating continuous functions on 𝐸, and 𝐒𝐎𝐂(𝐸) denotes the set of all cluster points of 𝐒𝐎𝐂(𝐸).

Proof. Let 𝑓 be any element in the closure of 𝐒𝐎𝐂(𝐸). Then there exists a sequence of points in 𝐒𝐎𝐂(𝐸) such that lim_𝑘→∞𝑓_𝑘=𝑓. To show that 𝑓 is slowly oscillating continuous, take any slowly oscillating sequence (𝑥_𝑛). Let 𝜀>0. Since (𝑓_𝑘) converges to 𝑓, there exists a positive integer 𝑁 such that for all 𝑥∈𝐸 and for all 𝑛≥𝑁, |𝑓(𝑥)−𝑓_𝑛(𝑥)|<𝜀/3. As 𝑓_𝑁 is slowly oscillating continuous, there exist a positive integer 𝑁₁≥𝑁 and a 𝛿 such that |𝑓_𝑁(𝑥_𝑛)−𝑓_𝑁(𝑥_𝑚)|<𝜀/3 if 𝑛≥𝑁₁ and 𝑛≤𝑚≤(1+𝛿)𝑛. Hence, for all 𝑛≥𝑁₁ and 𝑛≤𝑚≤(1+𝛿)𝑛, |𝑓(𝑥_𝑛)−𝑓(𝑥_𝑚)|≤|𝑓(𝑥_𝑛)−𝑓_𝑁(𝑥_𝑛)|+|𝑓_𝑁(𝑥_𝑛)−𝑓_𝑁(𝑥_𝑚)|+|𝑓_𝑁(𝑥_𝑚)−𝑓(𝑥_𝑚≤𝜀)|3+𝜀3+𝜀3=𝜀. (2.1) This completes the proof of the theorem.

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

Proof. The proof follows from Theorem 2.1.

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

Theorem 2.3. Let 𝐴 be a slowly oscillating compact subset of 𝐑 and let 𝑓∶𝐴→𝐑 be slowly oscillating continuous on 𝐴. Then 𝑓 is uniformly continuous on 𝐴.

Proof. Assume that 𝑓 is not uniformly continuous on 𝐴. Then there exist 𝜖₀ and sequences (𝑥_𝑛) and (𝑦_𝑛) in 𝐴 such that |𝑥_𝑛−𝑦_𝑛|<1/𝑛,|𝑓(𝑥_𝑛)−𝑓(𝑦_𝑛)|≥𝜖₀ (2.2) for all 𝑛∈𝐍. Since 𝐴 is slowly oscillating compact, there is a slowly oscillating subsequence (𝑥_𝑛𝑘) of (𝑥_𝑛). It is clear that the corresponding sequence (𝑦_𝑛𝑘) is also slowly oscillating, since |𝑦_𝑛𝑘−𝑦_𝑛𝑚|≤|𝑦_𝑛𝑘−𝑥_𝑛𝑘|+|𝑥_𝑛𝑘−𝑥_𝑛𝑚|+|𝑥_𝑛𝑚−𝑦_𝑛𝑚|. (2.3) If 𝑓 is slowly oscillating continuous, then the sequences (𝑓(𝑥_𝑛𝑘)) and (𝑓(𝑦_𝑛𝑘)) are slowly oscillating. By (2.2), this is not possible. Consequently, we obtain that if 𝑓 is slowly oscillating continuous on a slowly oscillating compact set 𝐴 of 𝐑, then 𝑓 is uniformly continuous on 𝐴.

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

3. Quasi slowly Oscillating Continuity

A sequence 𝐱=(𝑥_𝑛) is called quasi slowly oscillating, denoted by 𝐱∈𝐐𝐒𝐎, if (Δ𝑥_𝑛)=(𝑥_𝑛−𝑥_𝑛+1) is a slowly oscillating sequence. The concept of slowly oscillating continuity [4] suggests a new kind of continuity. A function 𝑓 is quasi slowly oscillating continuous if it transforms quasi slowly oscillating sequences to quasi slowly oscillating sequences, that is, (𝑓(𝑥_𝑛)) is quasi slowly oscillating whenever (𝑥_𝑛) is quasi slowly oscillating.

Notice that any slowly oscillating sequence is quasi slowly oscillating, but the converse is not always true. Indeed, let (𝑥_𝑛) be slowly oscillating. Hence, from the definition of slow oscillation of a sequence and the line below Δ𝑥_𝑛−Δ𝑥_𝑚=(𝑥_𝑛−𝑥_𝑚)−(𝑥_𝑛+1−𝑥_𝑚+1), (3.1) we immediately have that (𝑥_𝑛) is quasi slowly oscillating. To see that the converse is not true, we consider the following example. The sequence (𝑥_𝑛) defined by 𝑥_𝑛=∑^𝑛_𝑘=1((1/𝑘)∑^𝑘_𝑗=1(1/𝑗)) 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 𝑓 and 𝑔 defined by 𝑓(𝑥)=𝑥 and 𝑔(𝑥)=𝑥, the function 𝑓𝑔 is not quasi slowly oscillating. When both 𝑓 and 𝑔 are bounded and quasi slowly oscillating continuous, we have the following theorem.

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

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 𝐑. (𝐐𝐒𝐎-𝐐𝐒𝐎)(𝑥_𝑛)∈𝐐𝐒𝐎⇒(𝑓(𝑥_𝑛))∈𝐐𝐒𝐎. (𝐐𝐒𝐎-𝐜)(𝑥_𝑛)∈𝐐𝐒𝐎⇒(𝑓(𝑥_𝑛))∈𝐜.(c-c) For each 𝑥₀ in the domain of 𝑓, lim_𝑛→∞𝑓(𝑥_𝑛)=𝑓(𝑥₀) whenever lim_𝑛→∞𝑥_𝑛=𝑥₀. (𝐜-𝐐𝐒𝐎)(𝑥_𝑛)∈𝐜⇒(𝑓(𝑥_𝑛))∈𝐐𝐒𝐎.(𝐐𝐒𝐎-𝐒𝐎)(𝑥_𝑛)∈𝐐𝐒𝐎⇒(𝑓(𝑥_𝑛))∈𝐒𝐎. (𝐒𝐎-𝐐𝐒𝐎)(𝑥_𝑛)∈𝐒𝐎⇒(𝑓(𝑥_𝑛))∈𝐐𝐒𝐎. (𝐮) is uniform continuity of 𝑓.

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 𝑓 is quasi slowly oscillating continuous on 𝐑, then it is continuous in the ordinary sense.

Proof. Let (𝑥_𝑛) be any convergent sequence with lim_𝑘→∞𝑥_𝑘=𝑥₀. Then the sequence (𝑥₁,𝑥₀,𝑥₂,𝑥₀,…,𝑥_𝑛,𝑥₀,…) also converges to 𝑥₀ and is quasi slowly oscillating. By the hypothesis, the sequence (𝑓(𝑥₁)−𝑓(𝑥₀),𝑓(𝑥₀)−𝑓(𝑥₂),𝑓(𝑥₂)−𝑓(𝑥₀),𝑓(𝑥₀)−𝑓(𝑥₃),…) is slowly oscillating. It follows from the definition of slow oscillation that lim_𝑘→∞(𝑓(𝑥_𝑘)+𝑓(𝑥_𝑘+1))=2𝑓(𝑥₀) and lim_𝑘→∞(𝑓(𝑥_𝑘)−𝑓(𝑥_𝑘+1))=0. Hence, we have lim_𝑘→∞𝑓(𝑥_𝑘)=𝑓(𝑥₀). 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 (𝑓_𝑛) is a sequence of quasi slowly oscillating continuous functions defined on a subset 𝐸 of 𝐑 and let (𝑓_𝑛) is uniformly convergent to a function 𝑓, then 𝑓 is quasi slowly oscillating continuous on 𝐸.

Proof. Let (𝑓_𝑛) be a sequence of quasi slowly oscillating continuous functions defined on a subset 𝐸 of 𝐑 and (𝑓_𝑛) be uniformly convergent to a function 𝑓. Let (𝑥_𝑛) be a quasi slowly oscillating sequence and 𝜀>0. As (𝑓_𝑛) is uniformly convergent to 𝑓, there exists a positive integer 𝑁 such that |𝑓_𝑛(𝑥)−𝑓(𝑥)|<𝜀/5 for all 𝑥∈𝐸 whenever 𝑛≥𝑁. Since 𝑓_𝑁 is quasi slowly oscillating continuous, there exist a 𝛿>0 and a positive integer 𝑁₁ such that |Δ𝑓_𝑁(𝑥_𝑚)−Δ𝑓_𝑁(𝑥_𝑛𝜀)|<5 (3.2) for 𝑛≥𝑁₁(𝜀) and 𝑛≤𝑚≤(1+𝛿)𝑛. Now for 𝑛≥𝑁₁(𝜀) and 𝑛≤𝑚≤(1+𝛿)𝑛, we have |Δ𝑓(𝑥_𝑚)−Δ𝑓(𝑥_𝑛)|≤|𝑓(𝑥_𝑚)−𝑓_𝑁(𝑥_𝑚)|+|𝑓_𝑁(𝑥_𝑚+1)−𝑓(𝑥_𝑚+1)|+|𝑓_𝑁(𝑥_𝑛)−𝑓(𝑥_𝑛)|+|𝑓(𝑥_𝑛+1)−𝑓_𝑁(𝑥_𝑛+1)|+|𝑓_𝑁(𝑥_𝑚)−𝑓_𝑁(𝑥_𝑚+1)−𝑓_𝑁(𝑥_𝑛)+𝑓_𝑁(𝑥_𝑛+1≤𝜀)|5+𝜀5+𝜀5+𝜀5+𝜀5=𝜀. (3.3) This completes the proof of the theorem.

Definition 3.6. A subset 𝐹 of 𝐑 is called slowly oscillating compact [4] if whenever 𝐱=(𝑥_𝑛) is a sequence of points in 𝐹, there is a slowly oscillating subsequence 𝐲=(𝑦_𝑛𝑘) of 𝐱.

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

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

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

Notice that the union of two quasi slowly oscillating compact subsets of 𝐑 is quasi slowly oscillating compact. We see that any finite union of quasi slowly oscillating compact subsets of 𝐑 is quasi slowly oscillating compact, but any union of quasi slowly oscillating compact subsets of 𝐑 is not always quasi slowly oscillating compact. For example, consider the sets 𝐷_𝑛={𝑛} for 𝑛∈𝐍. Then for each constant 𝑛∈𝐍, the set 𝐷_𝑛 is quasi slowly oscillating compact, but ⋃^∞_𝑛=1𝐷_𝑛=𝐍 is not quasi slowly oscillating compact.

Theorem 3.7. Let 𝐹 be a quasi slowly oscillating compact subset of 𝐑 and let 𝑓 be a quasi slowly oscillating continuous function. Then 𝑓(𝐹) is quasi slowly oscillating compact.

Proof. Let 𝐲=(𝑦_𝑛) be any sequence of points in 𝑓(𝐹). Then there exists a sequence 𝐱=(𝑥_𝑛) such that 𝑦_𝑛=𝑓(𝑥_𝑛) for each 𝑛∈𝐍. As 𝐹 is quasi slowly oscillating compact, there exists a quasi slowly oscillating subsequence 𝐳=(𝑧_𝑘)=(𝑥_𝑛𝑘) of the sequence 𝐱. Since 𝑓 is quasi slowly oscillating continuous, 𝑓(𝐳)=(𝑓(𝑧_𝑘))=(𝑓(𝑥_𝑛𝑘)) is quasi slowly oscillating. It follows that 𝑓(𝐹) is quasi slowly oscillating compact.

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

Theorem 3.9. The set of quasi slowly oscillating continuous functions on a subset 𝐸 of 𝐑 is a closed subset of the set of all continuous functions on 𝐸, that is, 𝐐𝐒𝐎𝐂(𝐸)=𝐒𝐎𝐂(𝐸), where 𝐐𝐒𝐎𝐂(𝐸) is the set of all slowly oscillating continuous functions on 𝐸, and 𝐐𝐒𝐎𝐂(𝐸) denotes the set of all cluster points of 𝐐𝐒𝐎𝐂(𝐸).

Proof. Let 𝑓 be any element in the closure of 𝐐𝐒𝐎𝐂(𝐸). Then there exists a sequence of points in 𝐐𝐒𝐎𝐂(𝐸) such that lim_𝑘→∞𝑓_𝑘=𝑓. To show that 𝑓 is quasi slowly oscillating continuous, take any quasi slowly oscillating sequence (𝑥_𝑛). Let 𝜀>0. Since (𝑓_𝑘) converges to 𝑓, there exists a positive integer 𝑁 such that for all 𝑥∈𝐸 and for all 𝑘≥𝑁, |𝑓(𝑥)−𝑓_𝑘(𝑥)|<𝜀/5. As 𝑓_𝑁 is quasi slowly oscillating continuous, there exist a positive integer 𝑁₁≥𝑁 and a 𝛿 such that |Δ𝑓_𝑁(𝑥_𝑛)−Δ𝑓_𝑁(𝑥_𝑚)|<𝜀/5 if 𝑛≥𝑁₁ and 𝑛≤𝑚≤(1+𝛿)𝑛. Hence, for all 𝑛≥𝑁₁, |Δ𝑓(𝑥_𝑛)−Δ𝑓(𝑥_𝑚)|≤|Δ𝑓(𝑥_𝑛)−Δ𝑓_𝑁(𝑥_𝑛)|+|Δ𝑓_𝑁(𝑥_𝑛)−Δ𝑓_𝑁(𝑥_𝑚)|+|Δ𝑓_𝑁(𝑥_𝑚)−Δ𝑓(𝑥_𝑚)|=|𝑓(𝑥_𝑛)−𝑓(𝑥_𝑛+1)−𝑓_𝑁(𝑥_𝑛)+𝑓_𝑁(𝑥_𝑛+1)|+|Δ𝑓_𝑁(𝑥_𝑛)−Δ𝑓_𝑁(𝑥_𝑚)|+|𝑓_𝑁(𝑥_𝑚)−𝑓_𝑁(𝑥_𝑚+1)−𝑓(𝑥_𝑚)+𝑓(𝑥_𝑚+1)|≤|𝑓(𝑥_𝑛)−𝑓_𝑁(𝑥_𝑛)|+|𝑓_𝑁(𝑥_𝑛+1)−𝑓(𝑥_𝑛+1)|+|Δ𝑓_𝑁(𝑥_𝑛)−Δ𝑓_𝑁(𝑥_𝑚)|+|𝑓_𝑁(𝑥_𝑚)−𝑓(𝑥_𝑚)|+|𝑓(𝑥_𝑚+1)−𝑓_𝑁(𝑥_𝑚+1≤𝜀)|5+𝜀5+𝜀5+𝜀5+𝜀5=𝜀. (3.4) 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 ℎ, such that ℎ⁻¹ is Δ-pseudo continuous, and ℎ commutes the mappings at each point 𝑥.