As a refined notion, impulsive-mode controllability is suggested for the general class of linear continuous descriptor systems. This notion reveals rich properties of the general descriptor systems in terms of admissibility of initial conditions, admissibility of feedback control, and impulsive-mode elimination. The condition for impulsive-mode controllability is derived and proved to be necessary and sufficient for the existence of admissible feedback controls eliminating impulsive modes. Under impulsive-mode controllability, admissible feedback controls are constructed for eliminating impulsive modes, on the basis of numerically reliable decompositions of the system's coefficient matrices.