Baire*\ $1$, Darboux functions

Abstract

It is well known that a function $f:[0,1] \to R$ is Baire 1 if and only if in any closed set C there is a point ${x_0}$ at which the restricted function $f|C$ is continuous. Functions will be called Baire$^\ast$ 1 if they satisfy the following stronger property: For every closed set C there is an open interval (a, b) with $(a,b) \cap C \ne \emptyset$ such that $f|C$ is continuous on (a, b). Functions which are both Baire$^\ast$ 1 and Darboux are examined. It is known that approximately derivable functions are Baire$^\ast$ 1. Among other things it is shown here that ${L_p}$-smooth functions are Baire$^\ast$ 1. A new result about the ${L_p}$-differentiability of ${L_p}$-smooth, Darboux functions is shown to follow immediately from the main properties of Baire$^\ast$ 1, Darboux functions.