SIMULTANEOUS RECOVERY OF BAIRE ONE FUNCTIONS
Freiling, Vallin · Real Analysis Exchange · 1996
Given any countable collection of Baire one functions there is a single trajectory from which each of the functions is first return recoverable.[5] introduced the notion of first return paths to study differentiation properties of real functions.This launched a flurry of explorations.In one of the most intriguing of these, Darji, Evans, and O'Malley [2] (see also [1]) propose a creative and exciting new way to characterize Baire-one functions.They show that the function values can be described by a limiting process which is closely associated with continuity.This process is called "first return recovery".We consider here the possibility of simultaneously recovering countably many such functions, and by doing so, provide an alternate viewpoint for the proof of the Darji, Evans, O'Malley Theorem. O'MalleyAny countable dense sequence of real numbers T = {t 1 , t 2 , . . .} will be called a "trajectory".For each real number x we say that "t i is in the path to x", denoted t i ∈ T (x), if and only if none of the elements {t j } j