Accumulation Points of Graphs of Baire-1 and Baire-2 Functions

TL;DR

This paper investigates the conditions under which a subset of the plane can be realized as the accumulation points of a Baire-1 or Baire-2 function's graph, providing characterizations for bounded and unbounded cases.

Contribution

It characterizes the accumulation sets of Baire-1 and Baire-2 functions, extending previous graph-based definitions of Baire class functions.

Findings

Any such accumulation set for a Baire-2 function can be realized by an appropriate function.

Separate characterizations are provided for bounded and unbounded functions.

The paper explores the analogous problem for Baire-1 functions.

Abstract

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 $[0,1]\times\mathbb{R}$. When can we find a function $f:[0,1]\rightarrow\mathbb{R}$ 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.