Approaches to effective semi-continuity of real functions

Abstract

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 semicomputable of type one if its hypograph hypo(f) := {(x, y) : f(x) > y & x ∈ dom(f)} is recursively enumerably open in dom(f) × ℝ; 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) × ℝ. 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.