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].
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Weihrauch, K., Zheng, X. (1997). Effectiveness of the global modulus of continuity on metric spaces. In: Moggi, E., Rosolini, G. (eds) Category Theory and Computer Science. CTCS 1997. Lecture Notes in Computer Science, vol 1290. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0026990
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DOI: https://doi.org/10.1007/BFb0026990
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