Real Hypercomputation and Continuity
<|reference_start|>Real Hypercomputation and Continuity: 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 also discontinuous f:R->R. More precisely the present work considers the following three super-Turing notions of real function computability:   * relativized computation; specifically given oracle access to the Halting Problem 0' or its jump 0'';   * encoding real input x and/or output y=f(x) in weaker ways also related to the Arithmetic Hierarchy;   * non-deterministic computation.   It turns out that any f:R->R computable in the first or second sense is still necessarily continuous whereas the third type of hypercomputation does provide the required power to evaluate for instance the discontinuous sign function.<|reference_end|>
@article{ziegler2005real,
    title={Real Hypercomputation and Continuity},
    author={Martin Ziegler},
    journal={pp.177-206 in Theory of Computing Systems vol.41 (2007)},
    year={2005},
    doi={10.1007/s00224-006-1343-6},
    archivePrefix={arXiv},
    eprint={cs/0508069},
    primaryClass={cs.LO cs.CC}
}