The aim of this paper is to introduce some new classes of sets and some new classes of continuity namely $e^{\ast}$-open sets, $(\mathcal{D},\mathcal{S})$-sets, $(\mathcal{DS},\mathcal{E})$-sets, $(\mathcal{D},\mathcal{S})^{\ast}$-sets, $(\mathcal{DS},\mathcal{E})^{\ast}$-sets, $e^{\ast}$-continuity, $(\mathcal{D},\mathcal{S})^{\ast}$-continuity and $(\mathcal{DS},\mathcal{E})^{\ast}$$-continuity. Properties of these new concepts are investigated. Moreover, some new decompositions of continuity are provided.