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For a Tychonoff space X, we denote by Cp(X) the space of all real-valued continuous functions on X with the topology of pointwise convergence. In this paper, we study the κ-Fréchet Urysohn property and the weak Fréchet Urysohn property of Cp(X). Our main results are that (1) Cp(X) is κ-Fréchet Urysohn iff X has property (κ) (i.e. every pairwise disjoint sequence of finite subsets of X has a strongly...