Before revision:  By building on recent results in topology for shapes in algebras U_i, the paper provides, (1) a general formulation of continuous rule applications based on the  continuity of the mappings that describe these rule applications, (2) conditions that a mapping must satisfy to be suitable for analyzing rule continuity , and (3) procedures for computing continuous topologies in a given computation .
After revision:  By building on recent results in topology for shapes in algebras U_i, the paper provides, (1) a general formulation of continuous rule applications based on the continuity analysis, which uses recent results on shape topology and continuous mappings. A characterization is provided that distinguishes the suitable mapping forms from those that are inherently discontinuous or practically inconsequential for continuity analysis. It is also shown that certain inherent properties of shape topologies and continuous mappings provide an effective method of computing topologies algorithmically .