Introduction to Functional Equations

Abstract

Functional equations are equations where the unknowns are functions. A well-known example of functional equation is the following Cauchy equation:

$$ \phi (x + y) = \phi (x) + \phi (y). $$

((3.1))

A function φ is a solution of this equation if, for any two values x and y, the application of φ to x + y equals the addition of the application of φ to x and to y. Therefore, the equation establishes conditions that functions φ have to satisfy. Typical solutions of this Cauchy equation are the functions φ(x) = αx for an arbitrary value for α.