TY - JOUR

T1 - Composite continuous path systems and differentiation

AU - Alikhani-Koopaei, 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.

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U2 - 10.14321/realanalexch.35.1.0031

DO - 10.14321/realanalexch.35.1.0031

M3 - Article

AN - SCOPUS:85032379827

SN - 0147-1937

VL - 35

SP - 31

EP - 42

JO - Real Analysis Exchange

JF - Real Analysis Exchange

IS - 1

ER -