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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)].

##### MSC:
 26A21 Classification of real functions; Baire classification of sets and functions