This paper investigates continuity properties of the two-block '1 optimization resulting from the optimal design of BIBO stability robustness for discrete time systems in the presence of coprime factor perturbations. In general, the two-block '1 optimal design might lack continuity properties, since it might have no ÿnite dimensional solution. However, this paper shows that for a suitable given truncated order, the two-block '1 suboptimal design is continuous as a map from the plant to the suboptimal closed loop solution, if the plant has no zeros on the unit circle and has a unique suboptimal solution. Furthermore, if the set of plants is compactness, we show that there exists a uniform bound on the degree of suboptimal solution for all plants in the set such that their deviation of suboptimal value from the optimal one have the same bound, and the two-block '1 suboptimal design is uniformly continuous. In particular, in the case in which the plant has nonunique solution, a procedure is also developed on how to choose a solution that preserves continuity. The results enable the stability of robust '1 adaptive scheme to hold in the presence of coprime factor perturbations.