Functions and Continuity

Abstract

The notion of function is fundeimental to both pure and apphed mathematics. To give the “posh” definition first, if A and B are non-empty sets, a function f from A into B (usually written f: A→B) is defined as a subset f of the Cartesian product A × B with the property that, for all x in A and all y ₁, y ₂ in B,

$$ (x,y_1 ) \in f and (x,y_2 ) \in f \Rightarrow y_1 = y_2 . $$

To put it anothe way, for every x in A, the domain of the function, there is a unique y in B such that (x, y)∈f. In practice we denote this unique y by f(x), and say that f(x) is the image of x under f, or the value of f at x. We shall sometimes want to refer to the domain A of f as dom f. The set B is sometimes called the codomain of f