The minimum support for a functional on Cb(X)

Let Cb(X) be the set of all real-valued bounded continuous functions on a space X. Let ξ be a (not necessary linear) functional on Cb(X). A closed subset S of X is called a support for ξ if ξ(ƒ) = ξ(g) holds for any pair (ƒ, g) of functions in Cb(X) such that ƒ¦s = g¦s. We study some properties of supports for continuous functionals and uniformly continuous functionals on Cb(X) with certain topologies.