Path Derived Numbers and Path Derivatives of Continuous Functions with Respect to Continuous Systems of Paths

Composite Continuous Path Systems and Differentiation

“…The notion of a continuous system of paths was introduced in [1], where the path system E = { E x : x ∈ R }, which consists of compact sets, is required to be continuous as a function from R into the metric space of compact subsets of R endowed with the Hausdorff metric. This notion leads to continuous path derivative of a real valued function defined on R. Several nice properties of continuous path derivatives were shown in [1,2], and [3]. The present paper extends the notion of continuous path system by using the idea behind composite differentiation.…”

“…As usual, in the normed space C[0, 1] of all continuous functions from [0, 1] into R with the maximum norm, a property is said to hold typically if it is satisfied by members of a residual subset. In [3], the index of a path derived number of a function was defined, and it was shown that typically a continuous function has no finite E−derived number with index less than one at each x in [0, 1] when E is a continuous system of paths. Let us define the terms used in the property.…”

“…In Section 4, we generalize the main results given in [1], in particular it is shown that the composite continuous path derivative of a continuous real valued function is an element of Baire class one on a dense open set, the extreme composite continuous path derivatives of Borel measurable functions and of continuous functions are respectively Lebesgue measurable and members of Baire class two. In Section 5, some results concerning the typical properties of composite continuous path derived numbers of continuous real valued functions are pre-sented that generalizes some of the results given in [3]. The definitions and results given here could be stated on the real line, however for simplicity we consider the interval [a, b] as the ambient space.…”

“…The number C(α, E, f, x) = inf{γ({x n }, x) : {x n } ∈ S(α, E, f, x)} is called the E−index at x of α, where α ∈ D(E, f, x). The E−index of α in [0, 1] is defined to be C(α, E, f ) = inf{C(α, E, f, x) : x ∈ [0, 1]}.With the aid of Lemma 4.4, the proof of Theorem 5.2 is reduced to the following theorem on continuous path systems which is Theorem 4 on page 361 of[3].Theorem 5.1. Let E = {E x : x ∈ [0, 1]} be a continuous system of paths on [0, 1].…”