Suppose that \(x\) and \(y\) are two continuous real variables, which we may suppose to be represented geometrically by distances \(A_{0}P = x\), \(B_{0}Q = y\) measured from fixed points \(A_{0}\), \(B_{0}\) along two straight lines \(\Lambda\), M. And let us suppose that the positions of the points \(P\) and \(Q\) are not independent, but connected by a relation which we can imagine to be expressed as a relation between \(x\) and \(y\): so that, when \(P\) and \(x\) are known, \(Q\) and \(y\) are also known. We might, for example, suppose that \(y = x\), or \(y = 2x\), or \(\frac{1}{2}x\), or \(x^{2} + 1\). In all of these cases the value of \(x\) determines that of \(y\). Or again, we might suppose that the relation between \(x\) and \(y\) is given, not by means of an explicit formula for \(y\) in terms of \(x\), but by means of a geometrical construction which enables us to determine \(Q\) when \(P\) is known.

In these circumstances \(y\) is said to be a *function* of \(x\). This notion of functional dependence of one variable upon another is perhaps the most important in the whole range of higher mathematics. In order to enable the reader to be certain that he understands it clearly, we shall, in this chapter, illustrate it by means of a large number of examples.

But before we proceed to do this, we must point out that the simple examples of functions mentioned above possess three characteristics which are by no means involved in the general idea of a function, viz.:

(1) \(y\) is determined *for every value of \(x\)*;

(2) to each value of \(x\) for which \(y\) is given corresponds *one and only one value of \(y\)*;

(3) the relation between \(x\) and \(y\) is expressed by means of *an analytical formula*, from which the value of \(y\) corresponding to a given value of \(x\) can be calculated by direct substitution of the latter.

It is indeed the case that these particular characteristics are possessed by many of the most important functions. But the consideration of the following examples will make it clear that they are by no means essential to a function. All that is essential is that there should be some relation between \(x\) and \(y\) such that to some values of \(x\) at any rate correspond values of \(y\).

- I borrow this instructive example from Prof. H. S. Carslaw’s
*Introduction to the Calculus.*↩︎

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