ð ‘ ð œƒ -Ward Continuity

A function ð ‘“ is continuous if and only if ð ‘“ preserves convergent sequences; that is, ( ð ‘“ ( ð ›¼ ð ‘› ) ) is a convergent sequence whenever ( ð ›¼ ð ‘› ) is convergent. The concept of ð ‘ ð œƒ -ward continuity is defined in the sense that a function ð ‘“ is ð ‘ ð œƒ -ward continuous if it preserves ð ‘ ð œƒ -quasi-Cauchy sequences; that is, ( ð ‘“ ( ð ›¼ ð ‘› ) ) is an ð ‘ ð œƒ -quasi-Cauchy sequence whenever ( ð ›¼ ð ‘› ) is ð ‘ ð œƒ -quasi-Cauchy. A sequence ( ð ›¼ 𠑘 ) of points in ð ‘ , the set of real numbers, is ð ‘ ð œƒ -quasi-Cauchy if l i m ð ‘Ÿ → ∞ ( 1 / â„Ž ð ‘Ÿ ) ∑ 𠑘 ∈ ð ¼ ð ‘Ÿ | Δ ð ›¼ 𠑘 | = 0 , where Δ ð ›¼ 𠑘 = ð ›¼ 𠑘 + 1 − ð ›¼ 𠑘 , ð ¼ ð ‘Ÿ = ( 𠑘 ð ‘Ÿ − 1 , 𠑘 ð ‘Ÿ ] , and 𠜃 = ( 𠑘 ð ‘Ÿ ) is a lacunary sequence, that is, an increasing sequence of positive integers such that 𠑘 0 = 0 and â„Ž ð ‘Ÿ ∶ 𠑘 ð ‘Ÿ − 𠑘 ð ‘Ÿ − 1 → ∞ . A new type compactness, namely, ð ‘ ð œƒ -ward compactness, is also, defined and some new results related to this kind of compactness are obtained.