A new characterization of Baire class 1 functions

TL;DR

This paper introduces a novel characterization of Baire class 1 functions on ultrametric spaces, linking pointwise limits of Lipschitz functions with classical Borel function stratifications.

Contribution

It establishes that Baire class 1 functions are precisely the pointwise limits of Lipschitz functions and connects Baire classes with uniform limits of Delta functions.

Findings

Baire class 1 functions are pointwise limits of Lipschitz functions.

Baire class functions correspond to uniform limits of Delta functions.

New characterization links classical stratifications of Borel functions.

Abstract

We give a new characterization of the Baire class 1 functions (defined on an ultrametric space) by proving that they are exactly the pointwise limits of sequences of full functions (which are particularly simple Lipschitz functions). Moreover we highlight the link between the two classical stratifications of the Borel functions by showing that the Baire class functions of some level are exactly those obtained as uniform limits of sequences of Delta functions (of a corresponding level).