The answer by corserine is somehow invalid here. The Hahn-Banach theorem states that there exists a linear operator in the dual space of $C(X)^*$ that separates the closed convex set and the point outside it. However, the dual of $C(X)^*$ is not necessarily the space of space of continuous functions, as asked in the question. Hence, a specific derivation is required, i.e., by specifying the exact topology.
Yun
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