Computability and Continuity on the Real Arithmetic Hierarchy and the Power of Type-2 Nondeterminism

The sometimes so-called Main Theorem of Recursive Analysis implies that any computable real function is necessarily continuous. We consider three relaxations of this common notion of real computability for the purpose of treating also discontinuous functions f : R → R: – non-deterministic computation; – relativized computation, specifically given access to oracles like ∅′ or ∅′′; – encoding input x ∈ R and/or output y = f(x) in weaker ways according to the Real Arithmetic Hierarchy. It turns out that, among these approaches, only the first one provides