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Approaches to Effective Semi-Continuity of Real Functions

✍ Scribed by Xizhong Zheng; Vasco Brattka; Klaus Weihrauch

Publisher
John Wiley and Sons
Year
1999
Tongue
English
Weight
984 KB
Volume
45
Category
Article
ISSN
0044-3050

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✦ Synopsis

Abstract

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.

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