Effectiveness of the global modulus of continuity on metric spaces
In the "delta-epsilon" - definition of continuity of a function f between metric spaces, the value of delta depends on x, epsilon and function f as well. This kind of dependence can be described by a function, so called, "global modulus of continuity", which maps the triple (f,x, epsilon) to corresponding delta. By a recent result of Repovs and Semenov (Proc. Int. Conf. Topol. (Trieste, 1993), G. Gentili (Ed.), Rent. Ist. Mat. Univ. Trieste, vol. 25, 1993, pp. 441-446), there is a continuous global modulus of continuity for the function space C(X, Y) of all continuous functions from a locally compact metric space X to an arbitrary metric space Y. Based on Weihrauch's framework on computable metric spaces (K. Weihrauch, Theoret. Comput. Sci. 113 (1993) 191-210), we show that there is a computable global modulus of continuity for C(X, Y), if X is an "effectively locally compact" metric space and Y is a computable metric space. The proof is a direct construction not depending on the proof of Repovs and Semenov. (C) 1999 Elsevier Science B.V. All rights reserved.
global modulus of continuity continuous function computable metric space