Brian S. Thomson

More on the fundamental theorem of calculus

In a note [1] in the MONTHLY, Botsko and Gosser point out that the standard version of the Fundamental Theorem of Calculus holds when the usual derivative is replaced by the right-hand derivative. We would like to point out that by making a slight alteration in the usual definition of the Riemann integral, we can obtain an integral for which the Fundamental Theorem of Calculus holds in full generality. We begin by recalling one of the common definitions of the Riemann integral. If P = {a = xo 0 there exists 8 > 0 such that if P is a partition of mesh less than 8 and if ti E [xi_, xi], then

Rethinking the Elementary Real Analysis Course

Finally, the reader will probably observe the conspicuous absence of the time-honored topic in calculus courses, the “Riemann integral.” It may well be suspected that, had it not been for its prestigious name, this would have been dropped long ago, for (with due reverence to Riemann’s genius) it is certainly quite clear to any working mathematician that nowadays such a “theory” has at best the importance of a mildly interesting exercise in the general theory of measure and integration. Only the stubborn conservatism of academic tradition could freeze it into a regular part of the curriculum, long after it had outlived its historical importance. Of course, it is perfectly feasible to limit the integration process to a category of functions which is large enough for all purposes of elementary analysis, but close enough to the continuous functions to dispense with any consideration drawn from measure theory; this is what we have done by defining only the integral of regulated functions. When one needs a more powerful tool there is no point in stopping halfway, and the general theory of (Lebesgue) integration is the only sensible answer.

CHAPTER 5 – Differentiation

This chapter focuses on the interaction between differentiation properties and covering properties. The fundamental theorem of the calculus, presented to elementary students, relates integration and differentiation as inverse processes. The fundamental theorem of calculus suggests some important problems in measure theory. There are essentially two basic operations taken relative to a derivation basis for any extended real-valued function. There is no assumption that the limit superior must exceed the limit inferior. The fundamental relationship between a derivation basis and its dual is expressed in the chapter, which shows the essential role the dual plays in the study of limits. In addition to the limit operation in a derivation basis, one requires also an operation that can be used to reconstruct the measure. Some connections between the limit operation and the variational measure are analyzed in the chapter. The differentiation under strong Vitali assumptions is elaborated in the chapter. It is found that the strong Vitali property can be proved in some settings by establishing a variant, which involves a proportional cover.

Some Metapsychological Aspects of Mathematics Teaching

Summary In this paper we present and discuss two basic ideas which have arisen in the researches of the American writer‐philosopher L. R. Hubbard into the field of study. These ideas are stated as simple assertions, followed by brief discussions of their applications. The assertions are: 1. The only reason a person gives up a study or becomes confused or unable to learn is that he or she has gone past a word or symbol or phrase that was not understood. 2. In order for a student to learn and retain the data of a subject it is necessary that there be for him a proper balance among the elements: mass(any actual physical universe mass which is present or referred to in the subject), significance(ideas, relations, meaning, etc. in the subject) and ’doingness’(any actual activity involved in the study). The purpose of this paper is not only to encourage wider use of these two principles, but also to present a viewpoint quite distinct from that presented in this Journal in 1971 by Yeshuran.1

A theory of integration

This paper presents an exposition of the ideas fundamental to a theory of integration which has been investigated by R. Henstock [3], [4] and [5] and later by E. J. McShane [6]. The emphasis of these authors has been on a Riemann-type definition of an integral which possesses Lebesgue-type limit theorems and in particular on the problems of defining such an integral for vector-valued functions. There is an underlying simplicity in this area which is obscured by a Riemannoriented approach. We present here, in what seems to be the simplest kind of setting, the basic ideas of that part of the theory which interacts with the measure-theoretic tradition. The generalized versions of Itenstock andMcShane can then be considered to expand this setting. The paper concludes with a brief application of the theory to the familiar problem of integration in locally compact spaces.

ON WEAK VITALI COVERING PROPERTIES

1and (T% to denote the union of the sets in the sequence %\ always % will denote such a finite sequence of sets and our covering properties will be expressed in terms of approximation properties of such functions = J jd[L indicates that 4> is a set function defined on 2ft 0 by writing, for each Meffi0, (M) = lMfdii. The upper derivâtes, the lower derivâtes, and the derivâtes D*, D* are defined at each point x of JR in the obvious manner using the filterbase

A Strong Kind of Riemann Integrability.

Summary The usual definition of the Riemann integral as a limit of Riemann sums can be strengthened to demand more of the function to be integrated. This super-Riemann integrability has interesting properties and provides an easy proof of a simple change of variables formula and a novel characterization of derivatives. This theory offers teachers and students of elementary integration theory a curious and illuminating detour from the usual Rieman integral.

POROSITY AND APPROXIMATE DERIVATIVES

Etude de la porosite et des derivees approchees, en relation avec les resultats de A. Khintchine et G.H. Sindalovskii

ON THE HENSTOCK STRONG VARIATIONAL INTEGRAL

The theory of integration in division spaces introduced by Henstock ([3], [4]) serves to unite and simplify much of the classical material on nonabsolute integration as well as to provide a new approach to Lebesgue integration. In this paper we sketch a simplified approach to the division space theory and show how it can lead rapidly to the standard Lebesgue-type theory without a substantial departure from the usual methods; some applications to integration in locally compact spaces are briefly developed. No attempt has been made to state the best possible or most general results obtainable: our attention is fixed throughout on the strong variational integral for functions with values in a normed linear space.