Different types of continuous functions and their applications

Examining a contemporary form of function reveals certain traits that facilitate the resolution of particular issues. As a result, in order to help with a variety of applications and solve a number of challenging problems, we generalize to β -continuity and almost weak-continuity and develop a novel class of functions entitled β -weak, β -strong, and almost β -weak continuity. This unique kind of function is provided certain characterizations and basic attributes. We analyze how comparable functions and β -weakly continuous functions vary and are similar. We also show how various forms of weak continuity relate to one another. Finally, the authors demonstrate that if a function has a regular range and a highly unconnected domain, then β -continuity is identical.