We study the Cauchy problem for a controlled differential system with a parameter which is an element of some metric space Ξ, containing phase constraints on the control. It is assumed that at the given time instants t_k,k = 1,2,…,p, the solution x is continuous from the left and suffers a discontinuity, the value of which is x(t_k+0)-x(t_k ), belongs to some non-empty compact set of the space R^n. The notions of an admissible pair of this controlled impulsive system are introduced. The questions of continuity of admissible pairs are considered. Definitions of a priori boundedness and a priori collective boundedness on a given set S×K (where S⊂R^n is a set of initial values, K⊂Ξ is a set of parameter values) of the set of phase trajectories are considered. It is proved that if at some point (x_0,ξ)∈R^n×Ξ the set of phase trajectories is a priori bounded, then it will be a priori bounded in some neighborhood of this point.