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Approaches to Effective Semi‐Continuity of Real Functions
Author(s) -
Zheng Xizhong,
Brattka Vasco,
Weihrauch Klaus
Publication year - 1999
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.19990450407
Subject(s) - computability , computable analysis , computable function , mathematics , type (biology) , epigraph , function (biology) , discrete mathematics , pure mathematics , combinatorics , mathematical analysis , ecology , evolutionary biology , biology
For semi‐continuous real functions we study different computability concepts defined via computability of epigraphs and hypographs. We call a real function f lower semi‐computable of type one, if its open hypograph hypo( f ) is recursively enumerably open in dom( f ) × ℝ; we call f lower semi‐computable of type two, if its closed epigraph Epi( f ) is recursively enumerably closed in dom( f ) × ℝ; we call f lower semi‐computable of type three, if Epi( f ) is recursively closed in dom( f ) × ℝ. We show that type one and type two semi‐computability are independent and that type three semi‐computability 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.