The evolution of the function concept

The abstract definition of a function — a law that associates a single image with each element of the domain — seems obvious today, but it is the fruit of more than a century and a half of reflection. Retracing its stages helps us understand why the modern definition is phrased exactly as it is, insisting on the correspondence rather than on a formula.

Leibniz coins the term

The word function first appears in the writings of Gottfried Wilhelm Leibniz around 1673. For Leibniz a “function” was a geometric quantity depending on the points of a curve: the length of a tangent, a normal, a subtangent. The term is thus born tied to geometry and the differential calculus, not yet to the idea of a correspondence between numbers. In his correspondence with Johann Bernoulli the concept is refined, and it is Bernoulli himself who, in 1718, proposes to call a function “a quantity composed in any manner whatsoever from a variable and constants”.

Euler defines it

It is Leonhard Euler who gives the concept its first systematic form. In the Introductio in analysin infinitorum (1748) he writes that

a function of a variable quantity is an analytic expression composed in any manner whatsoever from that variable quantity and from numbers or constant quantities.

Here a function is essentially a formula: an expression such as $x^2 + 1$, $\sin x$, $\log x$. Euler also introduces the notation $f(x)$ that we still use. Later, while studying the problem of the vibrating string, Euler himself realised that this definition was too narrow: a plucked string takes on a profile made of several pieces that no single analytic expression describes. In this way a broader idea of a function as a “freely drawn” curve began to take shape.

Dirichlet generalises it

The decisive step is due to Peter Gustav Lejeune Dirichlet, in 1837. While studying the convergence of Fourier series, he abandons entirely the idea that a function must be given by a formula and proposes the modern definition: $y$ is a function of $x$ on an interval if to each value of $x$ there corresponds a single value of $y$, however that correspondence is established — even in words, or case by case. To show how general the definition was, Dirichlet exhibited the celebrated function

$$D(x) = \begin{cases} 1 & \text{if } x \text{ is rational}, \\ 0 & \text{if } x \text{ is irrational}, \end{cases}$$

known today as the Dirichlet function: a perfectly well-defined function that no elementary expression can represent and whose graph is impossible to draw. From this point on, what matters is the correspondence, not the formula — exactly the spirit of the definition we use in this chapter.

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Topics: Functions and properties
Concepts: Function
People: Gottfried Leibniz · Leonhard Euler (Eulero) · Peter Gustav Lejeune Dirichlet