Some remarks on Baire’s grand theorem

Abstract

We provide a game theoretical proof of the fact that if f is a function from a zero-dimensional Polish space to $\mathbb N^{\mathbb N}$ that has a point of continuity when restricted to any non-empty compact subset, then f is of Baire class 1. We use this property of the restrictions to compact sets to give a generalisation of Baire’s grand theorem for functions of any Baire class.