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 computabil-ity for the purpose of treating also discontinuous functions f : R → R: 1. non-deterministic computation; 2. relativized computation, specifically given access to oracles like 0′ or 0″; 3. 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 the required power.
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