In this paper continuity properties of the set-valued map p -> B-p(mu(0)), p is an element of (1, +infinity), are considered where B-p (mu(0)) is the closed ball of the space L-p ([t(0), theta]; R-m) centered at the origin with radius mu(0). It is proved that the set-valued map p -> B-p(mu(0)), p is an element of (1, +infinity), is continuous. Applying obtained results, the attainable set of the nonlinear control system with integral constraint on the control is studied. The admissible control functions are chosen from B-p (mu(0)). It is shown that the attainable set of the system is continuous with respect to p. (c) 2007 Elsevier Inc. All rights reserved.