7 - Functions and Mappings

Summary

Functions defined on an abstract set

This chapter is concerned with the extension of the idea of continuity to a function defined on an abstract set and taking values in an abstract set, possibly the same set. We recall the definition of a function or mapping given in Chapter 1.

Let E₁ and E₂ be two non-empty sets. By a mapping f of E₁into E₂ we mean a relation which associates with each element a of E₁ a single well-defined element of E₂ denoted by f(a) and called the image of a under the mapping f. The mapping f of E₁ into E₂ is denoted by f: E₁ → E₂. The set E₁ is called the domain of the mapping; the set {f(a): α ∈ E₁ of all points of E₂ which are images of points of E₂ is called the range of f. The range may be a proper subset of E₂; it may even be a single point, in which case the mapping is called a constant mapping. A function is, by definition, one-valued.

Another way of picturing a mapping is a generalization of the graph of elementary calculus. Consider the Cartesian product E₁ × E₂ which is the set of all ordered pairs (x,y) where x ∈ E₁, y ∈ E₂ A mapping or function f: E₁ → E₂ is a particular sort of subset of E₁ × E₂; a subset with the property that, amongst the ordered pairs (x, y) which form the function, each point a of E₁ occurs once and once only.