Variations on strong lacunary quasi-Cauchy sequences

We introduce a new function space, namely the space of N α θ (p)-ward continuous functions, which turns out to be a closed subspace of the space of continuous functions. A real valued function f defined on a subset A of R, the set of real numbers, is N α θ (p)-ward continuous if it preserves N α θ (p)-quasi-Cauchy sequences, that is, (f (xn)) is an N α θ (p)-quasi-Cauchy sequence whenever (xn) is N α θ (p)-quasi-Cauchy sequence of points in A, where a sequence (xk ) of points in R is called N α θ (p)-quasi-Cauchy if { Mathematical Formulas } where ∆xk = xk+1 − xk for each positive integer k, p is a constant positive integer, α is a constant in [0, 1], Ir = (kr−1, kr ], and θ = (kr ) is a lacunary sequence, that is, an increasing sequence of positive integers such that k0 ≠ 0, and hr : kr − kr−1 → ∞. Some other function spaces are also investigated.