Summary: By the sometimes so-called Main Theorem of Recursive Analysis, every computable real function is necessarily continuous. We wonder whether and which kinds of hypercomputation allow for the effective evaluation of discontinuous \(f :\;{\mathbb R}\to{\mathbb R}\), too. More precisely the present work considers the following three super-Turing notions of real function computability:

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relativized computation; specifically given oracle access to the Halting Problem \(\emptyset'\) or its jump \(\emptyset''\);

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encoding input \(x\in{\mathbb R}\) and/or output \(y = f(x)\) in weaker ways also related to the Arithmetic Hierarchy;

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nondeterministic computation.

It turns out that any \(f:{\mathbb R}\to{\mathbb R}\) computable in the first or second sense is still necessarily continuous, whereas the third type of hypercomputation provides the required power to evaluate for instance the discontinuous Heaviside function.