Continuous extension of functions from countable sets

We give a characterization of countable discrete subspace A of a topological space X such that there exists a (linear) continuous mapping φ:Cp⁎(A)→Cp(X) with φ(y)|A=y for every y∈Cp⁎(A). Using this characterization we answer two questions of A. Arhangel'skii. Moreover, we introduce the notion of well-covered subset of a topological space and prove that for well-covered functionally closed subset A of a topological space X there exists a linear continuous mapping φ:Cp(A)→Cp(X) with φ(y)|A=y for every y∈Cp(A).