On Simultaneous Extension of Continuous Partial Functions

For a metric space X let Cvc(X) (that is, the set of all graphs of real-valued continuous functions with a compact domain in X) be equipped with the Hausdorff metric induced by the hyperspace of nonempty closed subsets of X×R. It is shown that there exists a continuous mapping Φ : Cvc(X) → Cb(X) satisfying the following conditions: (i) Φ(f)| dom f = f for all partial functions f. (ii) For every nonempty compact subset K of X, Φ|Cb(K) : Cb(K) → Cb(X) is a linear positive operator such that Φ(1K) = 1X .