Variations on strongly lacunary quasi Cauchy sequences

Abstract

We introduce a new function space, namely the space of Nθ (p)-ward continuous functions, which turns out to be a closed subspace of the space of continuous functions for each positive integer p. Nθα(p)-ward continuity is also introduced and investigated for any fixed 0 < α ≤ 1, and for any fixed positive integer p. A real valued function f defined on a subset A of R, the set of real numbers is Nθα(p)-ward continuous if it preserves Nθα(p)-quasi-Cauchy sequences, i.e. (f (xn)) is an Nθα(p)-quasi-Cauchy sequence whenever (xn) is Nθα(p)-quasi-Cauchy sequence of points in A, where a sequence (xk) of points in R is called Nθα(p)-quasi-Cauchy if limr→∞1hrα∑k∈Ir|Δxk|p=0, where Δxk = xk+1−xk for each positive integer k, p is a fixed positive integer, α is fixed in ]0, 1], Ir = (kr−1, kr], and θ = (kr) is a lacunary sequence, i.e. an increasing sequence of positive integers such that k0 ≠ 0, and hr: kr−kr−1 →∞.