Computability on Continuou, Lower Semi-continuous and Upper Semi-continuous Real Functions

In this paper we investigate continuous and upper and lower semi-continuous real functions in the framework of TTE, Type-2 Theory of EEectivity. First some basic facts about TTE are summarized. For each of the function spaces, we introduce several natural representations based on diierent intiuitive concepts of \eeectivity" 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 non-continuity. 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. 1 Preliminaries By f : A ! B we denote a partial function from A to B with domain dom(f) A. If A = dom(f), we write f : A ! B as usual. Let ! be the set f0; 1; g of natural numbers. For any nite alphabet , is the set of all nite words over and ! is the set of all innnite sequences over. More generally, will denote a 1 a 2 an for = a 1 a 2 : : : a n 2 ff; !g n. For a word or sequence x, x(i) denotes i-th symbol of x. If x 2 , then jxj denotes the length of x. For p 2 ! and n 2 !, p j n denotes the initial segment of p of length n, i.e., p j n = p(0)p(1) : : : p(n ? 1). For any x; y 2 and any p; q 2 ! with p = xyq, x is a preex of p (denoted by x p), y is a subword of p (denoted by y < p) and q is a suux of p (denoted by p q). In the follwing let be a nite alphabet which contains 0, 1 and all other symbols we will use later. We assume that the reader is familiar with computability on ! and , which is the topic of classical computability theory (see 11, 15]). We call an innnite sequence p 2 !