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. 
Martin Ziegler