This book is unconvincing. It purports to generalize the definitions of derivative and integral so that any real function can be differentiated and integrated. Surprisingly it succeeds, but at the cost of introducing a new type of number to express the values of these new things, and introducing indeterminacy: each real function potentially has a infinite number of hyperderivatives and hyperintegrals. The unconvincing part of the book is that it doesn’t provide any evidence that these generalizations are useful; the introductory Chapter 1 claims they are useful in physics problems, but does not give any examples.

Roughly the first half of the book is devoted to developing hypernumbers and extrafunctions. Hypernumbers are a generalization of the real numbers that allow infinite and oscillating values. The construction is fairly straightforward: a hypernumber is an equivalence class of sequences of reals; if the sequence converges, the hypernumber is identified with the real number that is the limit. The construction has some similarities with the infinite and infinitesimal numbers of non-standard analysis. Hypernumbers do not include infinitesimals, because a sequence that goes to 0 is identified with the number 0. Extrafunctions are hypernumber-valued functions of hypernumbers; the name “hyperfunction” was already taken, so extrafunction is used instead. There are a number of methods of extending a function on the reals to a function on the hypernumbers, but the book is not tied to any particular method and does not claim that all functions can be so extended.

The rest of the book defines partial hyperderivatives and partial hyperintegrals and develops some of their elementary properties. The definition of partial hyperderivative is intricate, but here is a simplified version. To form a partial hyperderivative at a point, consider a sequence of points that converges to that point, and form the difference quotient at each point in the sequence. The sequence of these difference quotients belongs to an equivalence class that is a hypernumber, and this hypernumber is defined as the hyperderivative. Different sequences of points may produce different partial hyperderivatives at the same point. If all the hypernumbers are the same and are identified with a real number, then that real number is the old-style derivative. In this theory there also is no derivative function; there’s a collection of partial hyperderivatives at every point, although we may define a non-unique function that matches a partial hyperderivative at each point.

The development of partial hyperintegrals is similar, but depends on a sequence of partitions and the corresponding sequence of values of Riemann sums. The development is based on the Henstock-Kurzweil integral rather than the Riemann integral.

Then the book stops, without showing us why these definitions are useful. I don’t remember anyone complaining that not enough functions are differentiable, so the new definition is a doubtful benefit. People do complain about not enough functions being (Riemann) integrable, and this has led to a number of alternative integrals such as the Lebesgue and the Henstock-Kurzweil (see, for example, the surveys A Garden of Integrals by Burk and Theories of Integration by Kurtz & Swartz). The weaknesses of the Riemann integral boil down to two areas: pointwise limits of functions not being integrable, and derivatives not being integrable (this is needed for the Fundamental Theorem of Calculus). The real strength of the Lebesgue integral is its convergence theorems, but the book doesn’t mention any analogs of these. There is a version of the Fundamental Theorem of Calculus for partial hyperintegrals (p. 123), which does appear to be completely general although each choice of partitions for the partial hyperintegral requires a choice of partial hyperderivative for every point. The integrable-derivative property is also a property of the Henstock-Kurzweil integral, and is inherited by the partial hyperintegrals.

Bottom line: The target audience is unclear, and the book stops abruptly without showing us why the subject is useful, but otherwise it is a reasonable exposition of an intricate and esoteric subject. The book’s claim (p. vi) that “The book may be used for enhancing traditional courses of calculus for undergraduates” is wildly exaggerated.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.