On the Computational Properties of Weak Continuity Notions

The properties of continuous functions are very well-studied in computability theory and related areas. As it happens, there are many decompositions of continuity, which take the form \( \text {continuity }\leftrightarrow \text { [weak continuity notion }\textsf {A} + \text { weak continuity notion }\textsf {B}], \) for certain spaces and where the weak continuity notions are generally independent. In this paper, we investigate the properties of some of these weak continuity notions in Kleene’s computability theory based on S1–S9. Interestingly, certain weak continuity notions can be analysed fully with rather modest means (Kleene’s quantifier \(\exists ^{2}\) ), while others can be analysed with powerful tools (Kleene’s quantifier \(\exists ^{3}\) ), but not with weaker oracles. In particular, finding the supremum on the unit interval is possible using \(\exists ^{2}\) for certain weak continuity notions, while for others the italicised operation is computable in \(\exists ^{3}\) but not in weaker oracles.