Journal of Convex Analysis 32 (2025), No. 1, 169--188
Copyright Heldermann Verlag 2025
McShane's Condition for the Modulus of Continuity
Gerald Beer
Department of Mathematics, California State University, Los Angeles, U.S.A.
gbeer@cslanet.calstatela.edu
Michael D. Rice
Department of Mathematics and Computer Science, Wesleyan University, Middletown, U.S.A.
mrice@wesleyan.edu
McShane considered real-valued uniformly continuous functions defined on a subset of a metric space whose modulus of continuity has an affine majorant, and showed that they had uniformly continuous extensions to the entire space. Later it was shown that the same condition on a real-valued uniformly continuous function was both necessary and sufficient for the function to lie in the uniform closure of the real-valued Lipschitz functions. The main purpose of this note is to display conditions that are equivalent to the existence of an affine majorant for the modulus of an arbitrary function between metric spaces, with an emphasis on internal conditions on the function itself. No continuity assumptions on the objective function are made. We also give a simple formula for the uniform distance from a function whose modulus has an affine majorant to the set of Lipschitz functions that is valid for an important class of target spaces.
Keywords: Modulus of continuity, affine function, Lipschitz function, Lipschitz for large distances, Lipschitz in the small, bounded in the small, concave function, subadditive function, coarse map.
MSC: 26A15, 26A16, 41A30; 54C20, 54C30, 54E35.
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