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 appearance of shapes. While the shapes are kept unanalyzed throughout a computation, their descriptions can be defined retrospectively based on how the rules are applied. An important outcome of this is that continuity for rules is not built-in but it is "fabricated" retrospectively to explain a computation as a continuous process. An aspect of continuity analysis that has not been addressed in the literature is how to decide which mapping forms to use to study the continuity of rule applications. This is addressed in this paper in a new approach to 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 .