The Set of Real Functions are Countable in Applied Mathematics, Algebra of the Functions and Their Classification

Abstract

The set of real functions is countable since the functions must be computable, i.e. there must be an algorithm for computing them. But the set of algorithms is countable. Uncomputable functions are useless, they do not exist in applied mathematics. The set of computable real numbers is also countable. Uncomputable numbers are useless. The definition of algebra of computable real functions is given and a classification of subalgebras with one-element bases is constructed. This classification is a classification of functions too. Algebras with multielement bases are fictitious, they are useless for classification of functions. All infinite sequences of inclusions of subalgebras are constructed.