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We present robust stability results for discrete-time nonlinear systems using certainty equivalence output feedback, particularly those that employ a model predictive control (MPC) formulation to generate the feedback control law. To this end, we discuss nominal robustness properties of general discrete-time nonlinear systems, including those that use discontinuous control laws in the feedback loop. This is important for systems employing MPC since the method can, and sometimes necessarily does, result in discontinuous control laws. Coupling assumptions of nominal robustness with certain uniform observability or detectability assumptions (for each of which we give an observer), we assert that, in particular, MPC is robustly globally asymptotically stabilizing when used in a certainty equivalence output feedback structure. Finally, we give an example to illuminate our results.