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)


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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