First-return limiting notions in real analysis. (English) Zbl 1067.26004

Evans, Michael J.; O’Malley, Richard J.

Real Anal. Exch. 29(2003-2004), No. 2, 503-530 (2004).

The basis of considerations in this paper is the notion trajectory. One way to produce a trajectory is connected with dense orbit of continuous functions.
M. J. Evans, R. J. O’Malley, P. Humke, U. B. Darji (and others) have considered a most common method of specifying a trajectory assigning an enumeration (or ordering) to a given countable and dense set. On the basis of this notion one can construct a lot of interesting definitions leading us to the considerations connected with Baire class one, Darboux property, integrability etc. This article contains the most important results connected with this topic. From the abstract: “this survey attempts to summarize the results obtained and to point toward possible future directions for exploration”.

Reviewer: Ryszard Pawlak (Łódź)

MSC:

26A21	Classification of real functions; Baire classification of sets and functions
26A24	Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems

Keywords:

trajectory; first return; continuity; Baire class one; approachability; recoverability; integration; polygonal approximation