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

26A21 Classification of real functions; Baire classification of sets and functions