𝑁𝜃-Ward Continuity

• Main Author: Huseyin Cakalli
• Format: Article
• Language: English
• Published: Hindawi Limited 2012-01-01
• Series: Abstract and Applied Analysis
• Online Access: http://dx.doi.org/10.1155/2012/680456

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.