On some characterizations of Baire class one functions and Baire class one like functions. (English) Zbl 1224.26016

Bąkowska, Anna; Pawlak, Ryszard J.

Tatra Mt. Math. Publ. 46, 91-106 (2010).

Let \(\mathcal S\) and \(\mathcal T\) be families of positive real valued functions defined on \(\mathbb R\). By \(\mathfrak {B}_{\mathcal S, \mathcal T}\), we denote the family of all functions \(f\colon \mathbb R\to \mathbb R\) such that, for each function \(\varepsilon \in \mathcal T\), there is a function \(\delta _{\varepsilon }^f\in \mathcal S\) such that, for each \(x, y\in \mathbb R\), we have that, if \(| x-y| <\min \{ \delta ^f_{\varepsilon }(x), \delta ^f_{\varepsilon }(y)\}\), then \(| f(x)-f(y)| <\min \{ \varepsilon (f(x)), \varepsilon (f(y))\}\). Following the authors, we write \(\mathcal {X}^+\) for positive functions in \(\mathcal X\) and \(\mathbb R^{\mathbb R}\), \(C\), lsc and Const for the family of all, continuous, lower semicontinuous and constant functions, respectively. In the paper, the authors investigate families \(\mathfrak {B}_{\mathcal S, \mathcal T}\) when \(\mathcal S\) and \(\mathcal T\) are any of the sets \(\mathbb R^{\mathbb R+}\), \(C^+\), lsc\(^+\) and Const\(^+\). If \(\mathcal S=\mathbb R^{\mathbb R+}\) and \(\mathcal T=\)Const\(^+\), we have an \(\varepsilon \)-\(\delta \)-characterization of Baire one functions by P-Y. Lee, W-K. Tang and D. Zhao [Proc. Am. Math. Soc. 129, No. 8, 2273–2275 (2001; Zbl 0970.26004)].

Reviewer: Ján Borsík (Košice)

MSC:

26A21

Classification of real functions; Baire classification of sets and functions

Keywords:

Baire one function; \(B_1^{**}\)-function; uniformly continuous function; lower semicontinuous function; weakly lower semicontinuous function; upper semicontinuous function