On simultaneous extension of continuous partial functions

For a metric space $X$ let ${\cal C}_{vc}(X)$ (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 $X\times {\mathbf {R}}.$ It is shown that there exists a continuous mapping $\Phi :{\cal C}_{vc}(X)\rightarrow {\cal C}_b(X)$ satisfying the following conditions: (i) $\Phi (f)\vert \operatorname {dom}f= f$ for all partial functions $f.$ (ii) For every nonempty compact subset $K$ of $X,$ $\Phi \vert{\cal C}_b(K):{\cal C}_b(K) \rightarrow {\cal C}_b(X)$ is a linear positive operator such that $\Phi (1_K)=1_X$.