During the last few decades E. S. Thomas, S. J. Agronsky, J. G. Ceder, and T. L. Pearson gave an equivalent definition of the real Baire class 1 functions by characterizing their graph. In this paper, using their results, we consider the following problem: let T be a given subset of [0, 1] × ℝ. When can we find a function f : [0, 1] → ℝ such that the accumulation points of its graph are exactly the points of T? We show that if such a function exists, we can choose it to be a Baire-2 function. We characterize the accumulation sets of bounded and not necessarily bounded functions separately. We also examine the similar question in the case of Baire-1 functions. Mathematical Reviews subject classification: Primary: 26A21; Secondary: 26A15 Key words: accumulation points, Baire-1 functions, Baire-2 functions
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