Classification of weak continuities and decomposition of continuity

We first introduce 20 weak forms of continuity, which are closely related to 5 known weak forms of continuity. Then we classify them into 9 groups and give 12 decompositions of continuity. 1. Introduction. Continuity is one of the most important concepts in mathematics. In order to find deep properties of continuity, many weak forms of continuity were introduced in the literature. For instance, we have Levine's weak continuity [2] and semicontinuity [3], M. K. Singal and A. R. Singal's almost continuity [6], Husain's almost continuity [1], α-continuity of Mashhour et al. [4], and many others. Each of the above forms of continuity is strictly weaker than continuity. Theoretically, for each weak form of continuity, there is another weak form of continuity such that both of them imply continuity. In this connection, there is one result [3] for the general case. A special case is discussed in [7]. In this note, we develop these results. We introduce 20 weak forms of continuity, which are closely related with the above-mentioned weak continuities. Then we classify them into 9 groups and give 12 decompositions of continuity .