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.

Abstract. Let (X, d) be a complete metric space. We prove that there is a continuous, linear, regular extension operator from the space C * b of all partial, continuous, real-valued, bounded functions with closed, bounded domains in X to the space C * (X) of all continuous, bounded, real-valued functions on X with the topology of uniform convergence on compact sets. This is a variant of a result of Kunzi and Shapiro for continuous functions with compact, variable domains.

Let (X, d) be a complete metric space. We prove that there is a continuous, linear extension operator from the space of all partial, continuous, bounded metrics with closed, bounded domains in X endowed with the Hausdorff metric topology to the space of all continuous, bounded, metrics on X with the topology of uniform convergence on compact sets. This is a variant of the result of Tymchatyn and Zarichnyi for continuous metrics defined on closed, variable domains in a compact metric space. We get a similar result for the case of continuous real-valued functions.

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