Composite continuous path systems and differentiation

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Abstract

The concept of composite differentiation was introduced by O'Malley and Weil to generalize approximate differentiation. The concept of con- tinuous 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 com- posite 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.

Original languageEnglish (US)
Pages (from-to)31-42
Number of pages12
JournalReal Analysis Exchange
Volume35
Issue number1
DOIs
StatePublished - Jan 1 2010

All Science Journal Classification (ASJC) codes

  • Analysis
  • Geometry and Topology

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