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A simpler proof for the \(\epsilon\)-\(\delta\) characterization of Baire class one functions. (English) Zbl 1321.26006

The authors proved that if \(X\) and \(Y\) are separable metric spaces, then the following statements are equivalent:
(1) \(f:X\to Y\) is Baire class one;
(2) For every \(\varepsilon>0\) there is a positive function \(\delta:X\to \mathbb R^+\) such that \[ d_X(x,y)<\min\{\delta(x),\delta(y)\}\implies d_Y(f(x),f(y))<\varepsilon \] for every \(x,y\in X\).
This \(\varepsilon\)-\(\delta\) characterization of Baire class one functions was firstly proved by P.-Y. Lee et al. [Proc. Am. Math. Soc. 129, No. 8, 2273–2275 (2001; Zbl 0970.26004)] for complete separable metric spaces.

MSC:

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

Citations:

Zbl 0970.26004
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