New set types, decomposition of continuity and Γ-J-continuity via local closure function

New set types are defined by using the local closure function. The properties and necessary comparisons of these set types are given. With the help of these new set types, new types of continuity are introduced. Well-known continuity and J-continuity are generalized. Moreover, the decompositions of well-known continuity and Γ-J-continuity are obtained by new continuity types.