### 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 language | English (US) |
---|---|

Pages (from-to) | 31-42 |

Number of pages | 12 |

Journal | Real Analysis Exchange |

Volume | 35 |

Issue number | 1 |

State | Published - 2010 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Geometry and Topology

### Cite this

*Real Analysis Exchange*,

*35*(1), 31-42.

}

*Real Analysis Exchange*, vol. 35, no. 1, pp. 31-42.

**Composite continuous path systems and differentiation.** / Alikhani-Koopae, Aliasghar.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Composite continuous path systems and differentiation

AU - Alikhani-Koopae, Aliasghar

PY - 2010

Y1 - 2010

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85032379827&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85032379827&partnerID=8YFLogxK

M3 - Article

VL - 35

SP - 31

EP - 42

JO - Real Analysis Exchange

JF - Real Analysis Exchange

SN - 0147-1937

IS - 1

ER -