## Real Analysis Exchange

- Real Anal. Exchange
- Volume 35, Number 1 (2009), 31-42.

### Composite Continuous Path Systems and Differentiation

#### Abstract

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.

#### Article information

**Source**

Real Anal. Exchange, Volume 35, Number 1 (2009), 31-42.

**Dates**

First available in Project Euclid: 27 April 2010

**Permanent link to this document**

https://projecteuclid.org/euclid.rae/1272376222

**Mathematical Reviews number (MathSciNet)**

MR2657286

**Zentralblatt MATH identifier**

1210.26007

**Subjects**

Primary: 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15] 26A21: Classification of real functions; Baire classification of sets and functions [See also 03E15, 28A05, 54C50, 54H05]

Secondary: 26A27: Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives

**Keywords**

derived numbers composite derivatives continuous path systems path derivatives typical continuous functions

#### Citation

Alikhani-Koopaei, Aliasghar. Composite Continuous Path Systems and Differentiation. Real Anal. Exchange 35 (2009), no. 1, 31--42. https://projecteuclid.org/euclid.rae/1272376222