-Ward Continuity

A function is continuous if and only if preserves convergent sequences; that is, (()) is a convergent sequence whenever () is convergent. The concept of -ward continuity is defined in the sense that a function is -ward continuous if it preserves -quasi-Cauchy sequences; that is, (()) is an -quasi-Cauchy sequence whenever () is -quasi-Cauchy. A sequence () of points in , the set of real numbers, is -quasi-Cauchy if lim→∞(1/ℎ)∑∈|Δ|=0, where Δ=+1−, =(−1,], and =() is a lacunary sequence, that is, an increasing sequence of positive integers such that 0=0 and ℎ∶−−1→∞. A new type compactness, namely, -ward compactness, is also, defined and some new results related to this kind of compactness are obtained