Functions of real variables

This book is based on lectures which have been given for some years to intending honours mathematics students in the middle of their under-graduate studies. It presumes some previous study of elementary calculus. At this stage a student’s existing knowledge needs to be placed on a more exact basis. The first part of Chapter 1 therefore gives an outline of those properties of real numbers on which calculus is based. A full discussion of a construction of real numbers and the establishment of their basic properties is a specialised topic and too lengthy to be included. The outline given includes a brief description of the difference between the rational number field and the real number fieldnbased on the ideas of Dedekind, but footnotes indicate how this can be omitted by those who require a shorter outline. As the student has some experience of calculus the aim has been to express as much as possible in forms applicable to functions of several variables and independent of the number of variables before dealing with results special to the case of one variable. In the early part of the book properties of the trigonometric, exponential and logarithmic functions are used to illustrate the theory. They are not used to develop any part of the theory until after a later stage in which it is shown how they can be defined and their properties established. The latter part of the book consists of a selection of topics developed as interesting applications of the methods of advanced calculus. The dividing line between real variable theory and complex variable theory is the introduction of differentiation and integration with respect to a complex variable.