The Arithmetic Hierarchy of Real Functions

Inspired by Zheng and Weihrauch's arithmetic(al) hierarchy of real numbers, we investigate a hierarchy of computability levels for functions from R -> R. In particular, we seek to capture the classes of estimable, lower semicomputable, and upper semicomputable functions, understand how they are connected, and whether they can be generalized to important larger classes. We establish the lattice structure of our hierarchy along with its properties under various analytical and algebraic operations that are useful for calculations, particularly function composition. We also draw a connection to the arithmetic hierarchy of sets of natural numbers, extending classical results such as Shoenfield's limit lemma by allowing oracles with infinite input. Our main contribution is to make real hypercomputation results of Ziegler and Brattka accessible to computer scientists who are not computable analysts, which has direct applications to areas such as algorithmic information theory, descriptive set theory, and scientific computing.