Computability
Theory of recursive functions and effective computability
Theory of recursive functions and effective computability
COCOON '97 Proceedings of the Third Annual International Conference on Computing and Combinatorics
Computability on Continuou, Lower Semi-continuous and Upper Semi-continuous Real Functions
COCOON '97 Proceedings of the Third Annual International Conference on Computing and Combinatorics
A Foundation for Computable Analysis
SOFSEM '97 Proceedings of the 24th Seminar on Current Trends in Theory and Practice of Informatics: Theory and Practice of Informatics
Effectiveness of the Global Modulus of Continuity on Metric Spaces
CTCS '97 Proceedings of the 7th International Conference on Category Theory and Computer Science
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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 & × ∈ 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.