We offer a new and simpler proof of a recent ϵ-δ characterization of Baire class one functions using a theorem by Henri Lebesgue. The proof is more elementary in the sense that it does not use the Baire Category Theorem. Furthermore, the proof requires only that the domain and range be separable metric spaces instead of Polish spaces.
Fenecios, Jonald P.; Cabral, Emmanuel A. A Simpler Proof for the ϵ-δ Characterization of Baire Class One Functions. Real Anal. Exchange 39 (2014), no. 2, 441--446. https://projecteuclid.org/euclid.rae/1435670005