Approaches to Effective Semi-Continuity of Real Functions

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.