[...]... objected whether an exposition devoted entirely to a single equation (Cauchy’s Functional Equation) and a single inequality (Jensen’s Inequality) deserves the name An introduction to the Theory of Functional Equations and Inequalities However, the Cauchy equation plays such a prominent role in the theory of functional equations that the title seemed appropriate Every adept of the theory of functional equations. .. be different from A, and hence of the form P (a) with an a ∈ A If a ∈ P2 , then P1 is an initial segment of (P2 , ) Indeed, if x ∈ P1 and y < x, then y < a and hence y ∈ P (a) = P1 And if a ∈ P2 , then P2 is an initial segment of (P1 , ) Indeed, then P2 = A and hence of / the form P (b) with a b ∈ A If x ∈ P2 and y < x, then y < b and y ∈ P2 Thus one of the sets P1 , P2 is similar to its initial segment... appropriate to speak about the two classes of functions together Even in such a large book it was impossible to cover the whole material pertinent to the theory of the Cauchy equation and Jensen’s inequality The exercises at the end of each chapter and various bibliographical hints will help the reader to pursue further his studies of the subject if he feels interested in further developments of the theory. .. A and int A Some special letters are used to denote particular sets of numbers And so N denotes the set of positive integers, whereas Z denotes the set of all integers Q stands for the set of all rational numbers, R for the set of all real numbers, and C for the set of all complex numbers The letter N is reserved to denote the dimension of the underlying space The end of every proof is marked by the. .. use of the trichotomy law for the cardinals, which, however, cannot be proved at this stage What we actually prove here is that the m is impossible This is sufficient to prove the remaining theorems inequality α of the present chapter, and then the trichotomy law for the cardinals follows from Theorem 1.4.1 and 1.7.1, and hence also condition (1.5.3) We define Ω to be the order type of the set M (ℵ0 ), and. .. consider some functional equations on groups or related algebraic structures We assume that the reader has a basic knowledge of the calculus, theory of Lebesgue’s measure and integral, algebra, topology and set theory However, for the convenience of the reader, in the first part of the book we present such fragments of those theories which are often left out from the university courses devoted to them Also,... Thus f is of type β and its range γ∈Γ(β) is X In this proof of Zermelo theorem we have used the Axiom of Choice This could not be avoided, the Axiom of Choice had to be used in the proof Actually the Zermelo theorem is equivalent to the Axiom of Choice, as results from the following Theorem 1.7.2 If every set can be well ordered, then for every set there exists a choice function Proof Let X be an arbitrary... the sign Other symbols are introduced in the text, and for the convenience of the reader they are gathered in an index at the end of the volume The book is divided in chapters, every chapter is divided into sections When referring to an earlier formula, we use a three digit notation: (X.Y.Z) means formula Z in section Y in Chapter X The same rule applies also to the numbering of theorems and lemmas... Set Theory Axiom 1.1.5 Axiom of Infinity There exists a collection A of sets which contains the empty set ∅ and for every X ∈ A there exists a Y ∈ A consisting of all the elements of X and X itself: ∅ ∈ A and for every X ∈ A there exists a Y ∈ A such that (x ∈ Y ) ⇔ (x ∈ X) or (x = X) Axiom 1.1.6 Axiom of Choice The cartesian product of a non-empty family of nonempty sets is non-empty: If A = ∅ and. .. axiom1 Therefore the axiom of choice will equally be treated with the remaining axioms of the set theory and no special mention will be made whenever it is used The primitive notions of the set theory are: set, belongs to (∈), and being a relation type (τ ) [ατ A, R means α is a relation type of A, R ; cf Axiom 8] The eight axioms read as follows Axiom 1.1.1 Axiom of Extension Two sets are equal if and . paying tribute to the memory of the highly respected teacher, the excellent mathematician and one of the most outstanding researchers of functional equations and inequalities. Debrecen, October 2008 Attila. deserves the name An introduction to the Theory of Functional Equations and Inequalities. However, the Cauchy equation plays such a prominent role in the theory of functional equations that the title. Every adept of the theory of functional equations should be acquainted with the theory of the Cauchy equation. And a sys- tematic exposition of the latter is still lacking in the mathematical