A simpler proof for the \(\epsilon\)-\(\delta\) characterization of Baire class one functions (Q2356311)

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 \textit{P.-Y. Lee} et al. [Proc. Am. Math. Soc. 129, No. 8, 2273--2275 (2001; Zbl 0970.26004)] for complete separable metric spaces.