A Variation on Statistical Ward Continuity

Abstract

A real valued function defined on a subset E of R, the set of real numbers, is ρ-statistically downward continuous if it preserves ρ-statistical downward quasi-Cauchy sequences of points in E, where a sequence (α k ) of real numbers is called ρ-statistically downward quasi-Cauchy if limn→∞ 1 ρn |{k ≤ n : ∆α k ≥ ε}| = 0 for every ε > 0, in which (ρn) is a non-decreasing sequence of positive real numbers tending to ∞ such that lim sup n ρn n < ∞, ∆ρn = O(1), and ∆α k = α k+1 − α k for each positive integer k. It turns out that a function is uniformly continuous if it is ρ-statistical downward continuous on an above bounded set. 1 hr k∈Ir |α k − L| = 0, where I r = (k r−1 , k r ], and k 0 = 0, h r := k r − k r−1 → ∞ as r → ∞ and θ = (k r )is an increasing sequence of positive integers (see also [45], and [46]). In [43] Fridy and Orhan introduced the concept of lacunary statistically convergence in the sense that a sequence (α k ) of points in R is called lacunary statistically convergent, or Date: September 27, 2018. 2010 Mathematics Subject Classification. Primary: 40A05; Secondaries: 40D25; 40A35; 40A30; 26A15.

A variation on arithmetic continuity

“…We also see that arithmetical compactness and closedness together coincide with not only compactness, but also statistical sequential compactness ( [10]), λ-statistical sequential compactness ( [24]), ρ-statistical ward compactness ( [4], lacunary statistical sequential compactness ( [12]), strongly lacunary sequential compactness ( [3]), Abel sequential compactness ( [11]), and I-sequential compactness for a nontrivial admissible ideal I ( [20], [16]). …”

“…The preceding corollary ensures that arithmetical continuity implies either of the following continuities; ordinary continuity, statistical continuity, λ-statistical continuity ( [24]), ρ-statistical continuity ( [4]), lacunary statistical continuity ( [5]), and I-sequential continuity for any non trivial admissible ideal I of N ( [20]). …”

“…Using the idea of continuity of a real function in terms of sequences in the sense that a function preserves a certain kind of sequences in the above manner, many kinds of continuities were introduced and investigated, not all but some of them we recall in the following: slowly oscillating continuity ( [7]), quasi-slowly oscillating continuity ( [18]), ward continuity ( [15]), δ-ward continuity ( [8]), statistical ward continuity ( [10]), λ-statistical ward continuity ( [24]), ρ-statistical ward continuity ( [4]), lacunary ward continuity ( [12], strongly lacunary ward continuity ( [3]), and which enabled some authors to obtain conditions on the domain of a function for some The purpose of this paper is to give a further investigation of the concept of arithmetic continuity of a real function given in [40], and prove interesting theorems.…”

On downward half Cauchy sequences

“…Using the idea of continuity of a real function in terms of sequences, many kinds of continuities were introduced and investigated, not all but some of them we recall in the following: slowly oscillating continuity ( [16], [58]), quasi-slowly oscillating continuity ( [29]), ward continuity ( [5], [21], [13]), p-ward continuity ( [23]), δ-ward continuity ( [17]), δ 2 -ward continuity ( [2], statistical ward continuity ( [19], [20]), λ-statistical ward continuity (( [36], [49]), ρ-statistical ward continuity ( [7]), arithmetic continuity ( [8]) strongly lacunary ward continuity ( [14], [31], [44], [43], and [44]), lacunary statistical ward continuity ( [27], [32], and [60]), downward statistical continuity ( [24]), lacunary statistical downward continuity ( [10]) which enabled some authors to obtain conditions on the domain of a function to be uniformly continuous in terms of sequences in the sense that a function preserves a certain kind of sequences (see for example [58,Theorem 6],[13, Theorem 1 and Theorem 2], [29,Theorem 2.3].…”

A variation on lacunary statistical quasi cauchy sequences

“…In particular, lim denotes the limit function lim = lim n n on the linear space c. On the other hand, Çakall¬ has introduced a generalization of compactness, a generalization of connectedness via a method of sequential convergence in [5] and [11], respectively. In recent years, using the same idea, many kinds of continuities were introduced and investigated, not all but some of them we state in the following: slowly oscillating continuity [6], ward continuity [7], -ward continuity [8], statistical ward continuity [9], lacunary statistical ward continuity [13], and -statistically ward continuity [12]. Investigation of some of these kinds of continuities lead some authors to …nd conditions on the domain of a function for some characterizations of uniform continuity of a real function in terms of sequences in the above manner(see [29,Theorem 8], [7,Theorem 7], and [3, Theorem 1]).…”