Hajrudin Fejzić

Peano path derivatives

In this paper we introduce Peano path derivatives as a natural extension of the notion of path derivatives. We give a sufficient condition on a system of paths to ensure the corresponding Peano path derivative is Baire 1. As consequences, we obtain that unilateral approximate and unilateral Iapproximate Peano derivatives are Baire one. In studying generalized derivatives, one of the first questions that is addressed is that of Baire classification of the derivative. For example, it is known that ordinary, approximate and Peano derivatives are Baire 1. In [1] the authors introduced path derivatives as a method for unifying approaches to proving properties of various derivatives. Their starting point is the following definition. Definition 1. Let x ∈ R. A path leading to x is a set Ex ⊂ R such that x ∈ Ex and x is a point of accumulation of Ex. A system of paths is a collection E = {Ex| x ∈ R} such that each Ex is a path leading to x. For f : R → R and a system of paths E = {Ex| x ∈ R} we say that f is E-differentiable at x if lim h→0, x+h∈Ex f(x+ h)− f(x) h exists. The limit is called the E-derivative of f. It was shown in [1] that if the system of paths has the property that whenever x < y are sufficiently close the sets Ex and Ey intersect in the intervals (2x− y, x) to the left of x and (y, 2y − x) to the right of y, then the E-derivative is Baire 1. The authors called this property the External Intersection Condition (EIC) and, by showing that several generalized derivatives are path derivatives with systems satisfying the EIC, they produced simple proofs that these derivatives are Baire 1. Due to the bilateral nature of the EIC the analogous results for unilateral generalized derivatives cannot be obtained this way. Also, this condition is not designed to handle higher order generalized derivatives such as Peano derivatives and their generalizations. Here we introduce Peano path derivatives as a natural extension of the notion of path derivatives. By modifying and generalizing the EIC to this new setting we give a sufficient condition on a system of paths to ensure the corresponding Peano path derivative is Baire 1. As one consequence, we obtain a new and simple proof that approximate Peano derivatives are Baire 1. In fact, by freeing our intersection condition of bilateral restraints, we even show that unilateral approximate and unilateral I-approximate Peano derivatives are Baire 1. Received by the editors October 25, 1995 and, in revised form, March 20, 1996. 1991 Mathematics Subject Classification. Primary 26A24; Secondary 26A21. c ©1997 American Mathematical Society 2651 2652 HAJRUDIN FEJZIĆ AND DAN RINNE Definition 2. Let f : R → R and let E = {Ex| x ∈ R} be a system of paths. We say that f is n-times PeanoE-differentiable at x if there are numbers f1(x), ..., fn(x) such that lim h→0, x+h∈Ex f(x+ h)− f(x)− hf1(x)− · · · − h n! fn(x) hn = 0.

A property of Peano derivatives in several variables

Let f be a function of several variables that is n times Peano differentiable. Andreas Fischer proved that if there is a number M such that fα ≥ M or fα ≤ M for each α, with |α| = n, then f is n times differentiable in the usual sense. Here that result is improved to permit the type of one-sided boundedness to depend on α.

A mean value theorem for generalized Riemann derivatives

Functional differences that lead to generalized Riemann derivatives were studied by Ash and Jones in (1987). They gave a partial answer as to when these differences satisfy an analog of the Mean Value Theorem. Here we give a complete classification.

Extending n times differentiable functions of several variables

AbstractIt is shown that n times Peano differentiable functions defined on a closed subset of

Infinite approximate Peano derivatives

\mathbb{R}^m

Convex functions and Schwarz derivatives

and satisfying a certain condition on that set can be extended to n times Peano differentiable functions defined on

Measure Zero Sets with Non-Measurable Sum

\mathbb{R}^m