Approaches to Effective Semi-continuity of Real Functions

Extended Abstract
  • Vasco Brattka
  • Klaus Weihrauch
  • Xizhong Zheng
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1449)


By means of different effectivities of the epigraphs and hypographs of real functions we introduce several effectivizations of the semi-continuous real functions. We call a real function f lower semi-computable of type one if its hypograph hypo(f): = (x, y): f(x) > y & x ∈ dom(f) is recursively enumerably open in dom(f) × IR; f is lower semi-computable of type two if its closed epigraph Epi(f): = (x, y): f(x) ≤ y & x ∈ dom(f) is recursively enumerably closed in dom(f) × IR and f is lower semi-computable of type three if Epi(f) is recursively closed in dom(f) × IR. These semi-computabilities and computability of real functions are compared. We show that, type one and type two semi-computability are independent and that type three semicomputability plus effectively uniform continuity implies computability which is false for type one and type two instead of type three. We show also that the integral of a type three semi-computable real function on a computable interval is not necessarily computable.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Vasco Brattka
    • 1
  • Klaus Weihrauch
    • 1
  • Xizhong Zheng
    • 1
  1. 1.Theoretische Informatik IFernUniversität HagenHagenGermany

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