We introduce the notions of upper/lower (τ, m)-J-continuous multifunctions and obtain many characterizations of such multifunctions. The notion is obtained from a multifunc-tion F :(X, τ ) → (Y, σ, J) and several generalizations of J-open sets on the ideal topological space (Y, σ, J). If F is single valued, m = σ and J = {∅}, then the above multifunction is a (τ, m)-continuous function.