On continuous extension of uniformly continuous functions and metrics

Abstract

Abstract. We prove that there exists a continuous regular, positive homogeneous extension operator for the family of all uniformly continuous bounded real-valued functions whose domains are closed subsets of a bounded metric space (X, d). In particular, this operator preserves Lipschitz functions. A similar result is obtained for partial metrics and ultrametrics.1. Introduction. The theory of extensions of continuous functions and (pseudo)metrics has a long history. The Tietze-Urysohn theorem asserts that every continuous real-valued function on a closed subset of a metric space X admits a continuous extension to X. Hausdorff [7] proved an analogous theorem for metrics. McShane [9] showed that every uniformly continuous real-valued function from a closed subset of a metric space X which admits a concave modulus function ϕ such that lim t→0 ϕ(t) = 0 has a uniformly continuous extension to X. It is easy to see that the above condition on the modulus function is necessary. The chief contribution of this paper is to show that McShane's technique may be modified to give a continuous extension operator for several classes of functions. We also use a modification of Bing's formula [4] to construct continuous operators extending uniformly continuous metrics and ultrametics defined on the family of closed subsets of a metric (ultrametric) space.Dugundji [5] proved that one could extend continuously all functions with a fixed domain.

Simultaneous extension of continuous and uniformly continuous functions

“…Finally, let us remark that the extension operators constructed in this paper are complementary to a result on simultaneous extension of bounded uniformly continuous functions obtained in [3,Theorem 3.1]. The method used in [3] is based on the original arguments of McShane for proving Theorem 1.3. Our extension operators can also be compared with a nonlinear extension operator constructed by Borsuk [9].…”

Simultaneous Extension of Continuous and Uniformly Continuous Functions

“…Furthermore, this construction gives at once a sublinear extension operator Φ : C * (A) → C * (X) which is an isotone isometry and preserves uniform continuity. Finally, let us remark that the extension operators constructed in this paper are complementary to a result for simultaneous extension of bounded uniformly continuous function obtained in [3,Theorem 3.1]. The method used in [3] is based on the original arguments of McShane for proving Theorem 1.3.…”

On continuous linear operators extending metrics

“…In particular, Kunzi and Shapiro [12] proved that there exists a continuous, regular, linear operator extending continuous, real functions defined on compact subsets of a metric space. A variant of the Kunzi-Shapiro result for the noncompact case was obtained in [11] (see also [5]). Tymchatyn and Zarichnyi obtained an analogue of the Kunzi-Shapiro theorem for (pseudo)metrics [15].…”