As the size of the bibliography (which contains only a fraction of the existing literature on functional equations) indicates, it would be hopeless to try to present in one chapter of this book an even approximately complete history of functional equations (in several variables). So we restrict ourselves to a few notes (cf. also Aczél 1966c, pp. 5–12; Dhombres 1986) on the beginnings of this part of mathematics, some milestones in its development and a sketchy panorama of its present aspirations and applications.

Definition of linear and quadratic functions by functional equations in the Middle Ages and application of an implied characterization by Galileo

The emergence of functional equations was necessarily connected to the development of the notion of function, but we cannot, of course, go into the details of that history here. Because of the absence of any notion of function it would be very contrived to interpret passages of Euclid or Archimedes as even disguised formulations of functional equations. One should also differentiate between stating, for given functions, properties which amount to functional equations satisfied by them, and determining all functions with such properties, that is, solving these functional equations.

An important historical role of functional equations has been the definition of functions by functional equations (or their paraphrases). But often it was not shown (though it was implied) that these functions are the only solutions of these equations.

This chapter is entirely devoted to conditional Cauchy equations which were introduced in Chapter 6 and already studied in Chapter 7. In the first section, by using a fixed point theorem and also the order structure of ℝ, and by adding a strong regularity condition on the unknown function, we deduce the addundancy of some conditions on rather thin sets. In the second and third sections, we use group decomposition and then proceed to an application to additive number theory. An analogous equation in the fourth section yields an application to information theory, for mean codeword lengths. In the two last sections, we come back to additive number theory, proving the basic result that logarithms are the only monotonic functions defined on integers which transform a product into a sum. We extend this and similar results in various ways.

Expansions of the Cauchy equation from curves

Thus far, for conditional Cauchy equations, we have been dealing with rather ‘large’ subsets Z (see Chapters 6 and 7). Under some additional regularity assumptions on f, we may proceed to far smaller Z. A typical result can be obtained in the plane by using for Z some curve (Dhombres 1979, pp. 3.32–3.37). We recall from Section 7.1 that Z ⊂ ℝ2 is addundant in a class of functions if the restriction of the Cauchy equation from ℝ2 to Z changes neither the domain nor the form of its general solution in the class.