## Real Analysis Exchange

- Real Anal. Exchange
- Volume 41, Number 2 (2016), 315-330.

### Accumulation Points of Graphs of Baire-1 and Baire-2 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 graphs. 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.

#### Article information

**Source**

Real Anal. Exchange, Volume 41, Number 2 (2016), 315-330.

**Dates**

First available in Project Euclid: 30 March 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.rae/1490839334

**Mathematical Reviews number (MathSciNet)**

MR3597323

**Zentralblatt MATH identifier**

06848932

**Subjects**

Primary: 26A21: Classification of real functions; Baire classification of sets and functions [See also 03E15, 28A05, 54C50, 54H05]

Secondary: 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) {For properties determined by Fourier coefficients, see 42A16; for those determined by approximation properties, see 41A25, 41A27}

**Keywords**

accumulation points Baire-1 functions Baire-2 functions

#### Citation

Maga, Balázs. Accumulation Points of Graphs of Baire-1 and Baire-2 Functions. Real Anal. Exchange 41 (2016), no. 2, 315--330. https://projecteuclid.org/euclid.rae/1490839334