Even though there are many deep connections between mathematical analysis and the mathematics of computation, these two fields are usually taught completely separately from each other. This traditional approach has the disadvantage that the concepts from analysis are introduced in a manner that is not really optimal for the theory of computational mathematics to be built upon, thus making it unnecessarily difficult for novices to understand the links between the fields and to see how ideas from one area can be exploited in the other. In order to resolve this issue, Römisch and Zeugmann have written this book in a very innovative way to emphasize the connections.
The book’s contents essentially cover the topics that one would expect to find in a beginner’s calculus textbook (real and complex numbers, metric and normed spaces, continuity, one-dimensional differential and integral calculus) and an introductory textbook on numerical analysis (approximation and interpolation, numerical differentiation and integration, and the numerical solution of differential and integral equations). What makes the book special is that the authors have included additional chapters dealing with topics that bridge the border between the two areas. This includes a chapter on ordinary differential equations but also, and probably more importantly, the basics of functional analysis and operator theory that are required to investigate and properly understand the fundamental concepts of numerical mathematics such as stability, consistency, and convergence. Moreover, the complete exposition is such that the transition from one field to the other is made as easy as possible.
The book is written in a nicely readable style. It comes with plenty of carefully designed figures where color has been used in a well-thought-out way. Moreover, the text contains a large number of problems and exercises and many remarks describing the historical developments that have led to the current state of the art. I fully recommend it to be used by (beginning and advanced) students as material for self-study or by lecturers as a textbook.