This Dover republication of the second edition of Lawrence Graves’ 1946 textbook first became available in 2009, giving the present day mathematics community access to an influential book — a book which is notable for its breadth and ambition. Though originally conceived as an introductory textbook for undergraduates, the passage of time and other factors make this a textbook best avoided by undergraduates and most beginning graduate students.

The book begins with a quick introduction to logic and a rigorous development of the real number system. After dispensing with foundational considerations, the author deals with topological properties of n-dimensional Euclidean space, a discussion that culminates in a proof of the Heine-Borel Theorem. This is probably the most dated portion of the book. Graves makes use of some topological notions while avoiding others and does not provide a unifying framework for the geometric ideas being presented. He also makes the unusual choice of working with the compactification of n-dimensional Euclidean space which includes the points at positive and negative infinity along every axis. This is bound to cause a certain amount of vertigo for readers who are unaccustomed to this setting. For instance, the statement of the Heine-Borel theorem looks strange when there is no mention of boundedness.

Once Graves has the prerequisite machinery in place, he launches into a whirlwind tour of the usual topics one finds in a real analysis course: limits, continuity, differentiation, the Riemann integral, and uniform convergence. This portion of the book is notable for its extremely fast pace and the generality with which he examines the limit concept (functions need not be single valued). The treatment of uniform convergence is also interesting: Graves considers uniformity with respect to any parameter/variable and proves some very general results which are due to his mentor E. H. Moore. The reader is obliged to learn these results thoroughly, as Graves squeezes an incredible amount of mileage out of them in succeeding chapters.

The remaining two-thirds of the book deals with a wide range of specialized topics. There are chapters devoted to implicitly defined functions, ordinary differential equations, the Lebesgue integral, the Stieltjes integral, transfinite numbers, and metric spaces. Graves’ discussion of implicitly defined functions is much deeper than what one usually finds in current textbooks, and it includes a version of the implicit function theorem that has no differentiability requirements. As for his treatment of the Lebesgue integral, Graves’ approach is that of Riesz: a measurable function is defined to be a limit of step functions. The coverage of the Lebesgue integral is extensive (two long chapters) and much less hurried than the rest of the book. Graves does an excellent job of covering the fundamentals of this particular topic in a way that avoids unnecessary abstractions.

The chapters on transfinite numbers and metric spaces were added in the second edition and seem somewhat out of place. It is unfortunate that first half of the book did not benefit from the improved notation and more modern looking topological arguments that are introduced in the final chapter.

My overall impression of the book is that it is well organized and largely free of errors. It is also far denser than most textbooks written with undergraduates in mind. There are a small number of exercises that occur at irregular intervals, and they struck me as largely an afterthought. On the other hand, the book is very strong when it comes to providing useful and important references. Anyone who wishes to understand what the study of real analysis was like during the early part of the twentieth century might do well to start with this book since Graves often cites the original author of a particular argument or example.

The book does have problems and, as one might expect, these problems are exacerbated by the amount of time which has passed since the book’s first printing. One major problem is the notation, which is very unusual and often difficult to read. For instance, Graves uses a strange system of dots in place of normal delimiters like parentheses; it is unfortunate that entire definitions are often stated using this formalism. Another problem with the book is that many theorems are stated in a generality which seems inconsistent with its stated mission of providing undergraduates with an introduction to real analysis. Arguments are often terse and formalistic, making it difficult to develop an intuitive feel for the subject. Finally, while the book does contain some excellent examples (often complete with bibliographic notes), they are far too few in number.

*The Theory of Functions of Real Variables* is an ambitious and important book which clearly influenced today’s authors. For evidence we need look no further than the bibliography of Walter Rudin’s highly regarded *Principles of Mathematical Analysis*. It is good that Dover is making this text available for a modern audience, as it definitely contains some interesting bits of mathematics that are seldom seen today. The book gives some insight into the development of real analysis as an undergraduate course, and it provides some useful bibliographic information. The overly general nature of much of the discussion, however, together with the dated and awkward notation and the dearth of examples and exercises means that it is not suitable as a textbook for a modern course.

Duane Farnsworth is an assistant professor at Clarion University of Pennsylvania. He did his graduate work at SUNY Buffalo where he completed his degree in 2007. His research interests include functional analysis and various types of applied mathematics. In his spare time he enjoys hiking, reading novels, and the periodic breaking and fixing of his computer.