Richard J. O'Malley

Universally Polygonally Approximable Functions

Abstract It has recently been established that any Baire class one function ƒ : [0, 1] → ℝ can be represented as the pointwise limit of a sequence of polygonal functions whose vertices lie on the graph of ƒ. Here we investigate the subclass of Baire class one functions having the additional property that for every dense subset D of [0, 1], the first coordinates of the vertices of the polygonal functions can be chosen from D.

UNIVERSALLY FIRST RETURN CONTINUOUS FUNCTIONS

It is known that the first return continuous functions are precisely the Darboux functions in Baire class 1, and that every such function can be changed via a homeomorphism into an approximately continuous function. Here we give two characterizations of the smaller class of universally first return continuous functions, one of which is the capacity of changing such a function via a homeomorphism into an approximately continuous function which is con- tinuous almost everywhere.

Selective, bi-selective, and composite differentiation

The real line is denoted by R and all functions will be realvalued and defined on R. The closure of a set A c R will be denoted by C1A. The selective and bi-selective derivatives have been studied by the first author in [2] and [6] respectively. The composite derivative is introduced here and we show that a composite derivative is a bi-selective derivative, determine when a composite derivative is a selective derivative, and find conditions under which a bi-selective derivative is a composite derivative. The paper is concluded with an example showing that sometimes selective and composite derivatives must be different.

SELECTIVE DIFFERENTIATION: REDEFINING SELECTIONS

The concept of selective differentiation was introduced and developed in [4]. Here we present selective differentiation from a different viewpoint or, rather, in a new framework. This framework takes some effort to erect but is profitable in terms of the results obtained. It originates from several very simple observations. First, any closed nondegenerate interval [a, b] can be considered as a point (a, b) in the upper half-plane H= {(x, y); x<y}. Second, most notions of a sequence of intervals I, =[an, bn] converging to a point x0 can be translated into an equivalent notion of the sequence (an, bn) in H converging to the point (xo, xo) of the boundary D. For example, [an, bn] w-converges to :co, using the definition in [5], ff and only if (an; bn) converges to (Xo, xo) inside some Stolz angle. Third; for any function f: R~R let G(x,y)= f(y)-f(x) , G: H-+R. Then the study of various differentiability properties off y--x

Irrational Rotations and Differentiability

In this paper we investigate first return differentiation when the trajectories are restricted to irrational rigid rotations.