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In high school algebra, we learn about algebraic equations involving one or more unknown real numbers. Functional equations are much like algebraic equations, except that the unknown quantities are functions rather than real numbers. This book is about functional equations: their role in contemporary mathematics as well as the body of techniques that is available for their solution. Functional equations appear quite regularly on mathematics competitions. So this book is intended as a toolkit of methods for students who wish to tackle competition problems involving functional equations at the high school or university level.

Let us begin by restating and solving Cauchy’s functional equation. Let f : R → R be a continuous function satisfying (2.1) $$ f\left( {x + y} \right) = f\left( x \right) + f\left( y \right) $$ for all real x and y . We show that there exists a real number a such that f(x) = ax for all x ∈ R .

In this chapter, we consider functional equations involving a single variable. We have already considered these equations in Chapter 1, in the work of Charles Babbage. It is now time to examine some of these equations in greater detail. We begin by considering a useful technique known as linearization, which Babbage used to solve for iterative roots. After that, we look at some examples of conjugacy equations, which also arise from attempts to linearize functions.

In this chapter, we consider sundry functional equations in which there is some specialized character to the problem which requires techniques that are not easily exported to other settings. The restrictions that we consider here are of various types and are typically much stronger than the assumptions of previous chapters. Finally, at the end of the chapter, we consider a functional equation whose solution requires some group theory.

We have discussed many functional equations in the previous chapters. However, the next functional equation you encounter may be completely different from any of these and may require radically different methods to solve. In that case, some real ingenuity may be necessary.

It would have been appropriate to mention Georg Hamel and his connection with the theory of functional equations in our historical survey of Chapter 1. However, the work of Hamel is technical, and best studied after a thorough analysis of Cauchy’s contributions such as in Chapter 2. I have seen fit to place Hamel’s work in this appendix because his contribution to the subject depends upon a highly speculative axiom in mathematics, namely the Axiom of Choice, which is not accepted by all mathematicians.

It is always tempting after working on a problem for a while to turn to the answers at the back. However, the reader is warned not to turn to quickly to this section if a problem is a hard nut to crack. Often, a problem which seems to be almost impossible on a first run through will yield its secrets after you put it aside for a little while.