Computability on continuous, lower semi-continuous and upper semi-continuous real functions

Abstract

In this paper we extend computability theory to the spaces of continuous, upper semi-continuous and lower semi-continuous real functions. We apply the framework of TTE, Type-2 Theory of Effectivity, where not only computable elements but also computable functions on the spaces can be considered. First some basic facts about TTE are summarized. For each of the function spaces, we introduce several natural representations based on different intuitive concepts of “effectivity” and prove their equivalence. Computability of several operations on the function spaces is investigated, among others limits, mappings to open sets, images of compact sets and preimages of open sets, maximum and minimum values. The positive results usually show computability in all arguments, negative results usually express discontinuity. Several of the problems have computable but not extensional solutions. Since computable functions map computable elements to computable elements, many previously known results on computability are obtained as simple corollaries.