Composite continuous path systems and differentiation

The concept of composite differentiation was introduced by O'Malley and Weil to generalize approximate differentiation. The concept of continuous path systems was introduced by us. This paper combines these concepts to introduce the notion of composite continuous path systems into differentiation theory. It is shown that a number of results that hold for composite differentiation and for continuous path differentiation also hold for composite continuous path differentiation. In particular, a composite continuous path derivative of a continuous function is a Baire class one function on some dense open set, and extreme composite continuous path derivatives of a continuous function are Baire class two functions. It is also shown that extreme composite continuous path derivatives of a Borel measurable function are Lebesgue measurable. Finally, for each composite continuous path system E, continuous functions typically do not have E-derived numbers with E-index less than one.

Files

Metadata

Work Title Composite continuous path systems and differentiation
Access
Open Access
Creators
  1. Aliasghar Alikhani-Koopaei
Keyword
  1. Composite derivatives
  2. Continuous path systems
  3. Derived numbers
  4. Path derivatives
  5. Typical continuous functions
License In Copyright (Rights Reserved)
Work Type Article
Publisher
  1. Real Analysis Exchange
Publication Date 2010
Related URLs
Deposited February 01, 2024

Versions

Analytics

Collections

This resource is currently not in any collection.

Work History

Version 1
published

  • Created
  • Added 11-Alikhani-Composite_continuous_path_...-1.pdf
  • Added Creator Aliasghar Alikhani-Koopaei
  • Published
  • Updated Keyword, Publisher Identifier (DOI), Publication Date Show Changes
    Keyword
    • Composite derivatives, Continuous path systems, Derived numbers, Path derivatives, Typical continuous functions
    Publisher Identifier (DOI)
    • https://doi.org/10.14321/realanalexch.35.1.0031
    Publication Date
    • 2010-01-01
    • 2010
  • Updated Related URLs Show Changes
    Related URLs
    • https://projecteuclid.org/journals/real-analysis-exchange/volume-35/issue-1/Composite-Continuous-Path-Systems-and-Differentiation/rae/1272376222.full?tab=ArticleLink
  • Updated