Smooth approximations by continuous choice-functions

Abstract

We explore the existence of rational-valued approximation processes by continuous functions of two variables, such that the output continuously depends of the imposed error-bound. To this sake we prove that the theory of densely ordered sets with generic predicates is ℵ 0 -categorical. A model of the theory and a particular continuous choice-function are constructed. This function transfers to all other models by the respective isomorphisms. If some common-sense conditions are fulfilled, the processes are computable. As a byproduct, other functions with surprising properties can be constructed.Keywords choice • densely ordered sets • order topology • continuous functions • roots • algebraic and transcendental numbers • ℵ 0 -categorical theories.