Continuous linear extension of functions

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.