Nθ-Ward Continuity

Abstract EN

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 limr→∞(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.