6. Spaces of Continuous Functions

This chapter is devoted to representations of continuous functions and to applications of the concepts introduced so far. In Sect. 6.1 we define and discuss several representations of spaces of continuous real functions, in particular, representations via names of realizing programs, the “compact-open” representations and the representations by uniform approximation with rational polygons. In Sect. 6.2 we prove computability of any standard operations on functions, closed, open and compact sets. In particular, we prove a computable version of Urysohn’s lemma for closed subsets of ℝn. Computability of zero-finding for real functions under various restrictions is discussed in Sect. 6.3. Sect. 6.4 is devoted to computability problems of differentiation and integration, and Sect. 6.5 contains some further results on analytic functions.