Historical Background: Function and Related Concepts

Mental constructs, such as the concept of function, change through history as scholars get a better grip on important ideas. The meanings of the words used to represent these concepts also change, and for the same reason. The following text is quoted from An Introduction to the History of Mathematics by Howard Eves; our pop-up notes are added.

The concept of function, like the notions of space and geometry, has undergone a marked evolution, and every student of mathematics encounters various refinements of this evolution as his studies progress from the elementary courses of high school into the more advanced and sophisticated courses of the college postgraduate level.

The history of the term function furnishes another interesting example of the tendency of mathematicians to generalize and extend their concepts. The word function, in its Latin equivalent, seems to have been introduced by Leibniz¹ in 1694, at first as a term to denote any quantity connected with a curve, the radius of curvature, and so on. Johann Bernoulli,² by 1718, had come to regard a function as any expression made up of a variable and some constants, and Euler,³ somewhat later, regarded a function as any equation or formula involving variables and constants. This latter idea is the notion of a function formed by most students of elementary mathematics courses. The Euler concept remained unchanged until Joseph Fourier (1768–1830) was led, in his investigations of heat flow, to consider so-called trigonometric series.⁴ These series involve a more general type of relationship between variables than had previously been studied, and, in an attempt to furnish a definition of function broad enough to encompass such relationships, Lejeune Dirichlet (1805–1859)⁵ arrived at the following formulation: A variable is a symbol that represents any one of a set of numbers; if two variables x and y are so related that whenever a value is assigned to x there is automatically assigned, by some rule or correspondence, a value to y, then we say y is a (single-valued) function of x. The variable x, to which values are assigned at will, is called the independent variable, and the variable y, whose values depend upon those of x, is called the dependent variable. The permissible values that x may assume constitute the domain of definition of the function, and the values taken on by y constitute the range of values of the function.

The student of mathematics usually meets the Dirichlet definition of function in his introductory course in calculus. The definition is a very broad one and does not imply anything regarding the possibility of expressing the relationship between x and y by some kind of analytic expression; it stresses the basic idea of a relationship between two sets of numbers.

So, you see, our purpose so far in this chapter has been to induce you to move from an eighteenth-century understanding of function (that of Euler) to a nineteenth-century understanding (that of Dirichlet) — but with some sense of purpose for doing so.