Nθ‐Ward Continuity

A function f is continuous if and only if f preserves convergent sequences; that is, (f(αn)) is a convergent sequence whenever (αn) is convergent. The concept of Nθ‐ward continuity is defined in the sense that a function f is Nθ‐ward continuous if it preserves Nθ‐quasi‐Cauchy sequences; that is, (f(αn)) is an Nθ‐quasi‐Cauchy sequence whenever (αn) is Nθ‐quasi‐Cauchy. A sequence (αk) of points in R, the set of real numbers, is Nθ‐quasi‐Cauchy if lim r→∞(1/hr)∑k∈Ir|Δαk|=0, where Δαk = αk+1 − αk, 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 → ∞. A new type compactness, namely, Nθ‐ward compactness, is also, defined and some new results related to this kind of compactness are obtained.