Functions and Continuity

Abstract

Informally, a function f is described by a formula or a rule which, for a given input (usually a real number x), determines uniquely an output (again typically a real number y). The input is supposed to be an element of a set A called the domain of the function, and the output belongs to a set B called the codomain. When this happens we write y = f (x) and f: A → B. Examples of functions of this sort with A = B = ℝ (the set of all real numbers) are given for instance by (i) y = f(x) = x² + 1, (ii) y = g(x) = 1 if x ≥ 0, = 0 if x < 0, (iii) y = h(x) = the smallest prime number ≥ x. Many more examples will be given in the first section of this chapter.