The term decentralized integral controllability (DIC) pertains to the existence of stable decentralized controllers with integral action that have closed-loop properties such as stable independent detuning. It is especially useful to select control structures Systematically at the early stage of control system design because the only information needed for DIC is the steady-state process gain matrix. Here, a necessary and sufficient condition conjectured in the literature is proved. The real structured singular value which can exploit realness of the controller gain is used to describe computable conditions for DIC, The primary usage of DIC is to eliminate unworkable pairings. For this, two other simple necessary conditions are proposed. Examples are given to illustrate the effectiveness of the proposed conditions for DIC.