In this brief paper, we study the value function in maximum hands-off control. Maximum hands-off control, also known as sparse control, is the L-0-optimal control among the feasible controls. Although the L-0 measure is discontinuous and non-convex, we prove that the value function, or the minimum L-0 norm of the control, is a continuous and strictly convex function of the initial state in the reachable set, under an assumption on the controlled plant model. We then extend the finite-horizon maximum hands-off control to model predictive control (MPC), and prove the recursive feasibility and the stability by using the continuity and convexity properties of the value function. (C) 2015 Elsevier Ltd. All rights reserved.