Let mIO(X) be the family of ⋆-open (resp. α-I-open, pre-I-open, semi-I-open, β-I-open, etc.) sets in an ideal topological space (X, τ, I). By using mIO(X), we introduce and investigate notions m-I-continuous multifunction F : I) → (Y, σ) mi⋆ -continuous σ, J). The notion -continuity is a generalization m-I-continuity i⋆-continuity [9].