In this paper, the notions of $\mathcal{DS}^*$-sets and $\mathcal{DS}^*$-continuous functions are introduced and their properties and their relationships with some other types of sets are investigated. Moreover, some new decompositions of continuous functions are obtained by using $\mathcal{DS}^*$-continuous functions, $\mathcal{DS}$-continuous functions and $\mathcal D$-continuous functions.