New Types of Continuities

and Applied Analysis 3 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 0 and sequences xn and yn in A such that ∣ xn − yn ∣ ∣ < 1/n, ∣ ∣f xn − f ( yn )∣ ∣ ≥ 0 2.2 for all n ∈ N. Since A is slowly oscillating compact, there is a slowly oscillating subsequence xnk of xn . It is clear that the corresponding sequence ynk is also slowly oscillating, since ∣ ynk − ynm ∣ ∣ ≤ ∣∣ynk − xnk ∣ ∣ |xnk − xnm | ∣ xnm − ynm ∣ ∣. 2.3 If f is slowly oscillating continuous, then the sequences f xnk and f ynk 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 xn is called quasi slowly oscillating, denoted by x ∈ QSO, if Δxn xn − xn 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 xn is quasi slowly oscillating whenever xn is quasi slowly oscillating. Notice that any slowly oscillating sequence is quasi slowly oscillating, but the converse is not always true. Indeed, let xn be slowly oscillating. Hence, from the definition of slow oscillation of a sequence and the line below Δxn −Δxm xn − xm − xn 1 − xm 1 , 3.1 we immediately have that xn is quasi slowly oscillating. To see that the converse is not true, we consider the following example. The sequence xn defined by xn ∑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 4 Abstract and Applied Analysis 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 xn ∈ QSO ⇒ f xn ∈ QSO. QSO-c xn ∈ QSO ⇒ f xn ∈ c. c-c For each x0 in the domain of f , limn→∞f xn f x0 whenever limn→∞xn x0. c-QSO xn ∈ c ⇒ f xn ∈ QSO. QSO-SO xn ∈ QSO ⇒ f xn ∈ SO. SO-QSO xn ∈ SO ⇒ f xn ∈ 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 SOQSO . Theorem 3.2. If f is quasi slowly oscillating continuous on R, then it is continuous in the ordinary sense. Proof. Let xn be any convergent sequence with limk→∞xk x0. Then the sequence x1, x0, x2, x0, . . . , xn, 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 limk→∞ f xk f xk 1 2f x0 and limk→∞ f xk − f xk 1 0. Hence, we have limk→∞f xk 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 fn is a sequence of quasi slowly oscillating continuous functions defined on a subset E of R and let fn is uniformly convergent to a function f , then f is quasi slowly oscillating continuous on E. Abstract and Applied Analysis 5and Applied Analysis 5 Proof. Let fn be a sequence of quasi slowly oscillating continuous functions defined on a subset E of R and fn be uniformly convergent to a function f . Let xn be a quasi slowly oscillating sequence and ε > 0. As fn is uniformly convergent to f , there exists a positive integer N such that |fn x − f x | < ε/5 for all x ∈ E whenever n ≥ N. Since fN is quasi slowly oscillating continuous, there exist a δ > 0 and a positive integer N1 such that ∣ ΔfN xm −ΔfN xn ∣ ∣ < ε 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 xm −Δf xn ∣ ∣ ≤ ∣∣f xm − fN xm ∣ ∣ ∣ fN xm 1 − f xm 1 ∣ ∣ ∣ fN xn − f xn ∣ ∣ ∣ ∣f xn 1 − fN xn 1 ∣ ∣ ∣ fN xm − fN xm 1 − fN xn fN xn 1 ∣ ∣ ≤ ε 5 ε 5 ε 5 ε 5 ε 5 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 xn is a sequence of points in F, there is a slowly oscillating subsequence y ynk 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 methodG, anyG-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 setsDn {n} for n ∈ N. Then for each constant n ∈ N, the setDn is quasi slowly oscillating compact, but ⋃∞ n 1 Dn 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 yn be any sequence of points in f F . Then there exists a sequence x xn such that yn f xn for each n ∈ N. As F is quasi slowly oscillating compact, there exists a quasi slowly oscillating subsequence z zk xnk of the sequence x. Since f is quasi slowly oscillating continuous, f z f zk f xnk 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. 6 Abstract and Applied Analysis 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 , whereQSOC E is the set of all slowly oscillating continuous functions on E, andQSOC E denotes the set of all cluster points of QSOC E . Proof. Let f be any element in the closure ofQSOC E . Then there exists a sequence of points in QSOC E such that limk→∞fk f . To show that f is quasi slowly oscillating continuous, take any quasi slowly oscillating sequence xn . Let ε > 0. Since fk converges to f , there exists a positive integer N such that for all x ∈ E and for all k ≥ N, |f x − fk x | < ε/5. As fN is quasi slowly oscillating continuous, there exist a positive integer N1 ≥ N and a δ such that |ΔfN xn −ΔfN xm | < ε/5 if n ≥ N1 and n ≤ m ≤ 1 δ n. Hence, for all n ≥ N1, ∣ ∣Δf xn −Δf xm ∣ ∣ ≤ ∣∣Δf xn −ΔfN xn ∣ ∣ ∣ ΔfN xn −ΔfN xm ∣ ∣ ∣ ΔfN xm −Δf xm ∣ ∣ ∣ ∣f xn − f xn 1 − fN xn fN xn 1 ∣ ∣ ∣ ΔfN xn −ΔfN xm ∣ ∣ ∣ fN xm − fN xm 1 − f xm f xm 1 ∣ ∣ ≤ ∣∣f xn − fN xn ∣ ∣ ∣ fN xn 1 − f xn 1 ∣ ∣ ∣ ΔfN xn −ΔfN xm ∣ ∣ ∣ fN xm − f xm ∣ ∣ ∣ ∣f xm 1 − fN xm 1 ∣ ∣ ≤ ε 5 ε 5 ε 5 ε 5 ε 5 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 continuoush, such that h−1 is Δ-pseudo continuous, and h commutes the mappings at each point x. References1 H. Fast, “Sur la convergence statistique,” Colloqium Mathematicum, vol. 2, pp. 241–244, 1951.2 F. Dik, M. Dik, and İ. Çanak, “Applications of subsequential Tauberian theory to classical Tauberiantheory,” Applied Mathematics Letters, vol. 20, no. 8, pp. 946–950, 2007.3 J. Connor and K.-G. Grosse-Erdmann, “Sequential definitions of continuity for real functions,” TheRocky Mountain Journal of Mathematics, vol. 33, no. 1, pp. 93–121, 2003.4 H. Çakalli, “Slowly oscillating continuity,” Abstract and Applied Analysis, vol. 2008, Article ID 485706, 5pages, 2008.5 H. Çakalli, “Sequential definitions of compactness,” Applied Mathematics Letters, vol. 21, no. 6, pp. 594–598, 2008.6 H. Çakalli and P. Das, “Fuzzy compactness via summability,” Applied Mathematics Letters, vol. 22, no.11, pp. 1665–1669, 2009. Submit your manuscripts athttp://www.hindawi.com OperationsResearchAdvances in Hindawi Publishing Corporationhttp://www.hindawi.comVolume 2013Hindawi Publishing Corporationhttp://www.hindawi.comVolume 2013Mathematical Problemsin Engineering Hindawi Publishing Corporationhttp://www.hindawi.comVolume 2013Abstract andApplied Analysis ISRNAppliedMathematics Hindawi Publishing Corporationhttp://www.hindawi.comVolume 2013Hindawi Publishing Corporationhttp://www.hindawi.comVolume 2013International Journal ofCombinatorics Hindawi Publishing Corporationhttp://www.hindawi.comVolume 2013Journal of Function Spacesand Applications InternationalJournal ofMathematics andMathematicalSciences Hindawi Publishing Corporationhttp://www.hindawi.comVolume 2013