Text 3 functions of real variables
Suppose
that x and y are two continuous real variables, which we may suppose
to be represented geometrically by distances.
= x, B0Q
= y measured from fixed points A0,
B0
along
two straight lines ZM. 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 ½ x or x2
+ 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 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 text, 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 to values of y.
Finite and Infinite Classes. Before we proceed further it is necessary to make a few remarks about certain ideas of an abstract and logical nature which are of constant occurrence in pure mathematics.
In the first place, the reader is probably familiar with the notion of a class. It is unnecessary to discuss here any logical difficulties which may be involved in the notion of «a class»: roughly speaking we may say that «a class» is the aggregate or collection of all the entities or objects which possess a certain property, simple or complex. Thus we have the class of British subjects, or members of Parliament, or positive integers, or real numbers.
Moreover, the reader has probably an idea of what is meant by a finite or infinite class. Thus the class of British subjects is a finite class: the aggregate of all British subjects, past, present, and future, has a finite number n, though of course we cannot tell at present the actual value of n. The class of present British subjects has a number n which could be ascertained by counting, were the methods of the census effective enough.
On the other hand the class of positive integers is not finite but infinite. This may be expressed more precisely as follows. If n is any positive integer, such as 1,000, 1,000,000 or any number we like to think of, then there are more than n positive integers. Thus if the number we think of is 1,000,000, there are obviously at least 1,000,001 positive integers. Similarly the class of rational numbers, or of real numbers, is infinite. It is convenient to express this by saying that there are an infinite number of positive integers, or rational numbers, or real numbers. But the reader must be careful always to remember that by saying this we mean simply that the class in question has not a finite number of members such as 1,000 or 1,000,000...
The phrase “ n tends to infinity”. There is a somewhat different way of looking at the matter which it is natural to adopt. Suppose that n assumes successively the values 1, 2, 3,... The word «successively» naturally suggests, succession in time, and we may suppose n, if we like to assume these values at successive moments of time (e. g. at the beginnings of successive seconds). Then as the seconds pass n gets larger and larger and there is no limit to the extent of its increase. However large a number we may think of (e. g. 2, 147, 483, 647), a time will come when n has become larger than this number.
It is convenient to have a short phrase to express this unending growth of n, and we shall say that n tends to infinity, or n → ∞, this last symbol being usually employed as an abbreviation for «infinity». The phrase «tends to» like the word «successively» naturally suggests the idea of change in time, and it is convenient to think of the variation of n as accomplished in time in the manner described above. This, however, is a mere matter of convenience.
The variable n is a pure logical entity which has in itself nothing to do with time .
The reader cannot too strongly impress upon himself that when we say that n «tends to ∞» we mean simply that n is supposed to assume a series of values which increase continually and without limit.