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

Because the functions must be computable, that is, there must be a method for computing them, the set of real functions is countable. However, the number of algorithms may be counted. Uncomputable functions are useless in applied mathematics; they don’t exist. The set of countable real numbers is also a set of computable real numbers. Numbers that cannot be computed are useless. A classification of subalgebras with one-element bases is established, as well as a definition of algebra of computable real functions. This classification is also a function classification. Multielement bases algebras are fictitious, and thus are useless for classifying functions. Infinite sequences of subalgebra inclusions are built.

**Author (S) Details**

**M. A. Malkov
**Russian Research Center for Artificial Intelligence, Russia.

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