The design of key nonlinear systems often requires the use of expensive blackbox simulations presenting inherent discontinuities whose positions in the variable space cannot be analytically predicted. Without further precautions, the solution of related optimization problems leads to design configurations which may be close to discontinuities of the blackbox output functions. To account for possible changes of operating conditions, an acceptable solution must be away from unsafe regions of the space of variables. The objective of this work is to solve a constrained blackbox optimization problem with an additional constraint that the solution should be outside unknown zones of discontinuities or strong variations of the objective function or the constraints. The proposed approach is an extension of Mads and aims at building a series of inner approximations of these zones. The algorithm, called DiscoMads, relies on two main mechanisms: revealing discontinuities and progressively escaping the surrounding zones. A convergence analysis supports the algorithm and preserves the optimality conditions of Mads. Moreover, a stronger condition is derived by using the revelation mechanism. Numerical tests are conducted on analytical problems and on three engineering problems: the design of a simplified truss, the synthesis of a chemical component and the design of a turbomachine blade. The DiscoMads algorithm successfully solves these problems by providing a feasible solution away from discontinuous regions.