In this paper, we first discuss the stability of linearized error dynamics of the nonlinear observer used for time-continuous driving chaos synchronization and give the criteria on it. Then we find by theoretical analysis and numerical experiments that the observer can still synchronize with the original system under time-discrete driving provided that some conditions are met. Finally we derive the asymptotical stability criterion of the nonlinear observer used for time-discrete driving chaos synchronization. Simulations illustrate the validity of the criterion.