Effectiveness of the global modulus of continuity on metric spaces

  • Klaus Weihrauch
  • Xizhong Zheng
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1290)

Abstract

Let (X, dX) and (Y, dY) be metric spaces. By definition, there is a function h : (f, x, e) ↦ δ, (δ > 0), such that for all continuous function f : X → Y, x ∈ X and ∈ > 0: ∨x' ∈ X (dX (x, x′) < δ ⇒ dY (f (x), f (x′) < ∈). By a recent result of Repovš and Semenov [8], there is a function h continuous in f, x and e with this property, if (X, dX ) is locally compact. Based on Weihrauch's frameworks on computable metric space ([13]), we effectivize this result by showing that there is a computable function of this type. The proof is a direct construction not depending on [8].

Key words

Modulus of Continuity Metric Space Effective Analysis 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Klaus Weihrauch
    • 1
  • Xizhong Zheng
    • 1
  1. 1.Theoretische InformatikFern Universität HagenHagenGermany

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