Significant Modification of Pairwise-ω-continuous Functions with Associated Concepts

Abstract

The continuity is generalized by the notion of ω-continuous functions. In this research, we present a new weaker form for continuous functions called pairwise ω-continuous functions. Additionally, we define pairwise barely ω-continuous functions, a new, weaker form of barely ω-continuous functions. We study the basic characteristics and impacts of pairwise ω-continuous functions, clarifying their connection with typical continuity and providing perspectives on the wider field of topological analysis. It explores related ideas like the ω-limit, which describes how sequences behave over certain conditions when the function is applied. In addition, the concepts highlight the importance of pairwise ω-continuous functions in theoretical and practical conditions by discussing their relationships with other functional structures. An extensive number of demonstrative examples will be presented, along with the new results and theorems about pairwise barely ω-continuous and pairwise ω-continuous functions that generalize.