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 semi-computable of type one if its hypograph hypo(f): = (x, y): f(x) > y & x ∈ 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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
V. Brattka Computable Invariance. COCOON’97, Shanghai, China, August 1997.
X. Ge & A. Nerode Effective content of the calculus of variations I: semicontinuity and the chattering lemma. Annals of Pure and Applied Logic, 79 (1996), no. 1–3, 127–146.
X. Ge Private Communication, 1997.
A. Grzegorczy On the definitions of recursive real continuous functions. Fund. Math. 44 (1957), 61–71.
M. Pour-El & J. Richards Computability in Analysis and Physics. Springer-Verlag, Berlin, Heidelberg, 1989.
J. Ian Richards & Q. Zhou Computability of closed and open sets in Euclidean space, Preprint, 1992.
H. Jr. Rogers Theory of Recursive Functions and Effective Computability McGraw-Hill, Inc. New York, 1967.
K. Weihrauch Computability. EATCS Monographs on Theoretical Computer Secience Vol. 9, Springer-Verlag, Berlin, Heidelberg, 1987.
K. Weihrauch A foundation for computable analysis. in Douglas S. Bridges, Cristian S. Calude, Jeremy Gibbons, Steves Reeves and Ian H. Witten (editors), Combinatorics, Complexity, Logic, Proceedings of DMTCS’96, pp 66–89, Springer-Verlag, Sigapore, 1997.
K. Weihrauch & V. Brattka Computable Subsets of Euclidean Space. to appear in Theoret. Comput. Sci..
K. Weihrauch & X. Zheng Computability on Continuous, Lower Semi-Continuous and Upper Semi-Continuous Real Functions, to appear in Theoret. Comput. Sci. 211 (1998). (Extended abstract in Proc. of COCOON’97 Shanghai, China, August 1997, 166–175).
K. Weihrauch & X. Zheng Effectivity of the Global Modulus of Continuity on Metric Spaces, CTCS’97, Santa Margherita Ligure, Italy, September 1997, 210–219.
K. Weihrauch & X. Zheng A Finite Hierarchy of Recursively enumerable real numbers, submitted.
Q. Zhou Computability of closed and open sets in Euclidean space, Math. Log. Quart. 42 (1996), 379–409.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Brattka, V., Weihrauch, K., Zheng, X. (1998). Approaches to Effective Semi-continuity of Real Functions. In: Hsu, WL., Kao, MY. (eds) Computing and Combinatorics. COCOON 1998. Lecture Notes in Computer Science, vol 1449. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-68535-9_22
Download citation
DOI: https://doi.org/10.1007/3-540-68535-9_22
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-64824-6
Online ISBN: 978-3-540-68535-7
eBook Packages: Springer Book Archive