Step Functions and Product Rules

Step functions, the backbone of engineering analysis, have been somewhat neglected by mathematicians because of a serious inconvenience. In contrast to "good" functions, they are widely perceived as not conforming to product rules. The difficulty is augmented by ambiguous practice about the point of discontinuity of a step function. If two values are given to the stepping point, the step function ceases to be a function in a strictly mathematical sense. Some practitioners, consequently, defined the value of the stepping point from below; others from above; still others as an average value. These ambiguities further dampened mathematical ardor for step functions even as engineers continue using them with great success. Nevertheless, step and impulse functions deserve keen mathematical reflection, not in the least because they are intimately connected a subject close to the heart of mathematics: − the axiom of choice. The reader of this book will gain an appreciation of the axiom as a topological notion. The present work, by removing ambiguities and by examining closely the intricate relationship between step functions and product rules, uncovers a tight and unified mathematical theory that promises to revive mathematical ardor for the subject. Step functions under multiplication are seen to form a semi-group which under addition, are promoted into the ring of stair functions. This book thus becomes an integral part of a proper theory of functions. This book also forms Appendix 1 of the author's Theoretical Physics: The First Problem. The material proved so useful there in examining discontinuities, boundaries, and much else besides, that it was decided to make it available expeditiously and so separately.