## 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

Zbl 0970.26004
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