An Introduction to Mathematics · Alfred North Whitehead
The mathematical use of the term function
has been adopted also in common life. For example, "His temper is a function of his digestion," uses the term exactly in this mathematical sense. It means that a rule can be assigned which will tell you what his temper will be when you know how his digestion is working. Thus the idea of a "function" is simple enough, we only have to see how it is applied in mathematics to variable numbers. Let us think first of some concrete examples: If a train has been travelling at the rate of twenty miles per hour, the distance ( s miles) gone after any number of hours, say t , is given by s = 20 × t ; and s is called a function of t. Also 20 × t is the function of t with which s is identical. If John is one year older than Thomas, then, when Thomas is at any age of x years, John's age ( y years) is given by y = x + 1 ; and y is a function of x , namely, is the function x + 1.
In these examples t and x are called the
"arguments" of the functions in which they appear. Thus t is the argument of the function 20 × t , and x is the argument of the function x + 1. If s = 20 × t , and y = x + 1 , then s and y are called the "values" of the functions 20 × t and x + 1 respectively.
Coming now to the general case, we can define a function in mathematics as a correlation between two variable numbers, called respectively the argument and the value of the function, such that whatever value be assigned to the "argument of the function" the "value of the function" is definitely (i.e. uniquely) determined. The converse is not necessarily true, namely, that when the value of the function is determined the argument is also uniquely determined. Other functions of the argument x are y = x^2 ,
y = 2x^2 + 3x + 1 , y = x , y = log x , y = sin x. The last two functions of this group will be readily recognizable by those who understand a little algebra and trigonometry. It is not worth while to delay now for their explanation, as they are merely quoted for the sake of example.
Up to this point, though we have defined what we mean by a function in general, we have only mentioned a series of special functions. But mathematics, true to its general methods of procedure, symbolizes the general idea of any function. It does this by writing
F(x) , f(x) , g(x) , φ(x) , etc., for any function of x , where the argument x is placed in a bracket, and some letter like F , f , g , φ , etc., is prefixed to the bracket to stand for the function. This notation has its defects. Thus it obviously clashes with the convention that the single letters are to represent variable numbers; since here F , f , g , φ , etc., prefixed to a bracket stand for variable functions. It would be easy to give examples in which we can only trust to common sense and the context to see what is meant. One way of evading the confusion is by using Greek letters (e.g. φ as above) for functions; another way is to keep to f and F (the initial letter of function) for the functional letter, and, if other variable functions have to be symbolized, to take an adjacent letter like g.
With these explanations and cautions, we write y = f(x) , to denote that y is the value of some undetermined function of the argument x ; where f(x) may stand for anything such as x + 1 , x^2 - 2x + 1 , sin x , log x , or merely for x itself. The essential point is that when x is given, then y is thereby definitely determined. It is important to be quite clear as to the generality of this idea. Thus in y = f(x) , we may determine, if we choose, f(x) to mean that when x is an integer, f(x) is zero, and when x has any other value, f(x) is 1. Accordingly, putting y = f(x) , with this choice
for the meaning of f , y is either 0 or 1 according as the value of x is integral or otherwise. Thus f(1) = 0 , f(2) = 0 , f(2/3) = 1 , f(sqrt(2)) = 1 , and so on. This choice for the meaning of f(x) gives a perfectly good function of the argument x according to the general definition of a function.
A function, which after all is only a sort
of correlation between two variables, is represented like other correlations by a graph, that is in effect by the methods of coordinate geometry. For example, fig. 2 in Chapter II. is the graph of the function 1/v where v is the argument and p the value of the function. In this case the graph is only drawn for positive values of v , which are the only values possessing any meaning for the physical application considered in that chapter. Again in fig. 14 of Chapter IX. the whole length of the line AB , unlimited in both directions, is the graph of the function x + 1 , where x is the argument and y is the value of the function; and in the same figure the unlimited line A_1B is the graph of the function 1 - x , and the line LOL' is the graph of the function x , x being the argument and y the value of the function.
These functions, which are expressed by simple algebraic formulæ, are adapted for representation by graphs. But for some functions
this representation would be very misleading without a detailed explanation, or might even be impossible. Thus, consider the function mentioned above, which has the value 1 for all values of its argument x , except those which are integral, e.g. except for x = 0 , x = 1 , x = 2 , etc., when it has the value 0. Its appearance on a graph would be that of the straight line ABA' drawn parallel to the
axis XOX' at a distance from it of 1 unit of length. But the points, B , C_1 , C_2 , C_3 , C_4 , etc., corresponding to the values 0 , 1 , 2 , 3 , 4 , etc., of the argument x , are to be omitted, and instead of them the points O , B_1 , B_2 , B_3 , B_4 , etc., on the axis OX , are to be taken. It is easy to find functions for which the graphical representation is not only inconvenient but impossible. Functions which do not lend themselves to graphs are important in the
higher mathematics, but we need not concern ourselves further about them here.
The most important division between functions
is that between continuous and discontinuous functions. A function is continuous when its value only alters gradually for gradual alterations of the argument, and is discontinuous when it can alter its value by sudden jumps. Thus the two functions x + 1 and 1 - x , whose graphs are depicted as straight lines in fig. 14 of Chapter IX., are continuous functions, and so is the function 1/v , depicted in Chapter II., if we only think of positive values of v. But the function depicted in fig. 20 of this chapter is discontinuous since at the values x = 1 , x = 2 , etc., of its argument, its value gives sudden jumps.
Let us think of some examples of functions presented to us in nature, so as to get into our heads the real bearing of continuity and discontinuity. Consider a train in its journey along a railway line, say from Euston Station, the terminus in London of the London and North-Western Railway. Along the line in order lie the stations of Bletchley and Rugby. Let t be the number of hours which the train has been on its journey from Euston, and s be the number of miles passed over. Then s is a function of t , i.e. is the variable value corresponding to the variable argument t.
If we know the circumstances of the train's run, we know s as soon as any special value of t is given. Now, miracles apart, we may confidently assume that s is a continuous function of t. It is impossible to allow for the contingency that we can trace the train continuously from Euston to Bletchley, and that then, without any intervening time, however short, it should appear at Rugby. The idea is too fantastic to enter into our calculation: it contemplates possibilities not to be found outside the Arabian Nights; and even in those tales sheer discontinuity of motion hardly enters into the imagination, they do not dare to tax our credulity with anything more than very unusual speed. But unusual speed is no contradiction to the great law of continuity of motion which appears to hold in nature. Thus light moves at the rate of about 190,000 miles per second and comes to us from the sun in seven or eight minutes; but, in spite of this speed, its distance travelled is always a continuous function of the time.
It is not quite so obvious to us that the velocity of a body is invariably a continuous function of the time. Consider the train at any time t : it is moving with some definite velocity, say v miles per hour, where v is zero when the train is at rest in a station and is negative when the train is backing. Now we readily allow that v cannot change its
value suddenly for a big, heavy train. The train certainly cannot be running at forty miles per hour from 11.45 a.m. up to noon, and then suddenly, without any lapse of time, commence running at 50 miles per hour. We at once admit that the change of velocity will be a gradual process. But how about sudden blows of adequate magnitude? Suppose two trains collide; or, to take smaller objects, suppose a man kicks a football. It certainly appears to our sense as though the football began suddenly to move. Thus, in the case of velocity our senses do not revolt at the idea of its being a discontinuous function of the time, as they did at the idea of the train being instantaneously transported from Bletchley to Rugby. As a matter of fact, if the laws of motion, with their conception of mass, are true, there is no such thing as discontinuous velocity in nature. Anything that appears to our senses as discontinuous change of velocity must, according to them, be considered to be a case of gradual change which is too quick to be perceptible to us. It would be rash, however, to rush into the generalization that no discontinuous functions are presented to us in nature. A man who, trusting that the mean height of the land above sea-level between London and Paris was a continuous function of the distance from London, walked at night on Shakespeare's
Cliff by Dover in contemplation of the Milky Way, would be dead before he had had time to rearrange his ideas as to the necessity of caution in scientific conclusions.
It is very easy to find a discontinuous function, even if we confine ourselves to the
simplest of the algebraic formulæ. For example, take the function y = 1/x , which we have already considered in the form p = 1/v , where v was confined to positive values. But
now let x have any value, positive or negative. The graph of the function is exhibited in fig. 21. Suppose x to change continuously from a large negative value through a numerically decreasing set of negative values up to 0 , and thence through the series of increasing positive values. Accordingly, if a moving point, M , represents x on XOX' , M starts at the extreme left of the axis XOX' and successively moves through M_1 , M_2 , M_3 , M_4 , etc. The corresponding points on the function are P_1 , P_2 , P_3 , P_4 , etc. It is easy to see that there is a point of discontinuity at x = 0 , i.e. at the origin O. For the value of the function on the negative (left) side of the origin becomes endlessly great, but negative, and the function reappears on the positive (right) side as endlessly great but positive. Hence, however small we take the length M_2 M_3 , there is a finite jump between the values of the function at M_2 and M_3. Indeed, this case has the peculiarity that the smaller we take the length between M_2 and M_3 , so long as they enclose the origin, the bigger is the jump in value of the function between them. This graph brings out, what is also apparent in fig. 20 of this chapter, that for many functions the discontinuities only occur at isolated points, so that by restricting the values of the argument we obtain a continuous function for these remaining values. Thus it is evident
from fig. 21 that in y = 1/x , if we keep to positive values only and exclude the origin, we obtain a continuous function. Similarly the same function, if we keep to negative values only, excluding the origin, is continuous. Again the function which is graphed in fig. 20 is continuous between B and C_1 , and between C_1 and C_2 , and between C_2 and C_3 , and so on, always in each case excluding the end points. It is, however, easy to find functions such that their discontinuities occur at all points. For example, consider a function f(x) , such that when x is any fractional number f(x) = 1 , and when x is any incommensurable number f(x) = 2. This function is discontinuous at all points.
Finally, we will look a little more closely at the definition of continuity given above. We have said that a function is continuous when its value only alters gradually for gradual alterations of the argument, and is discontinuous when it can alter its value by sudden jumps. This is exactly the sort of definition which satisfied our mathematical forefathers and no longer satisfies modern mathematicians. It is worth while to spend some time over it; for when we understand the modern objections to it, we shall have gone a long way towards the understanding of the spirit of modern mathematics. The
whole difference between the older and the newer mathematics lies in the fact that vague half-metaphorical terms like "gradually" are no longer tolerated in its exact statements. Modern mathematics will only admit statements and definitions and arguments which exclusively employ the few simple ideas about number and magnitude and variables on which the science is founded. Of two numbers one can be greater or less than the other; and one can be such and such a multiple of the other; but there is no relation of "graduality" between two numbers, and hence the term is inadmissible. Now this may seem at first sight to be great pedantry. To this charge there are two answers. In the first place, during the first half of the nineteenth century it was found by some great mathematicians, especially Abel in
Sweden, and Weierstrass in Germany, that
large parts of mathematics as enunciated in the old happy-go-lucky manner were simply wrong. Macaulay in his essay on Bacon
contrasts the certainty of mathematics with the uncertainty of philosophy; and by way of a rhetorical example he says, "There has been no reaction against Taylor's theorem."
He could not have chosen a worse example. For, without having made an examination of English text-books on mathematics contemporary with the publication of this essay, the
assumption is a fairly safe one that Taylor's theorem was enunciated and proved wrongly in every one of them. Accordingly, the anxious precision of modern mathematics is necessary for accuracy. In the second place it is necessary for research. It makes for clearness of thought, and thence for boldness of thought and for fertility in trying new combinations of ideas. When the initial statements are vague and slipshod, at every subsequent stage of thought common sense has to step in to limit applications and to explain meanings. Now in creative thought common sense is a bad master. Its sole criterion for judgment is that the new ideas shall look like the old ones. In other words it can only act by suppressing originality.
In working our way towards the precise definition of continuity (as applied to functions) let us consider more closely the statement that there is no relation of "graduality" between numbers. It may be asked, Cannot one number be only slightly greater than another number, or in other words, cannot the difference between the two numbers be small? The whole point is that in the abstract, apart from some arbitrarily assumed application, there is no such thing as a great or a small number. A million miles is a small number of miles for an astronomer investigating the fixed stars, but a million
pounds is a large yearly income. Again, one-quarter is a large fraction of one's income to give away in charity, but is a small fraction of it to retain for private use. Examples can be accumulated indefinitely to show that great or small in any absolute sense have no abstract application to numbers. We can say of two numbers that one is greater or smaller than another, but not without specification of particular circumstances that any one number is great or small. Our task therefore is to define continuity without any mention of a "small" or "gradual" change in value of the function.
In order to do this we will give names to some ideas, which will also be useful when we come to consider limits and the differential calculus.
An "interval" of values of the argument x
of a function f(x) is all the values lying between some two values of the argument. For example, the interval between x = 1 and x = 2 consists of all the values which x can take lying between 1 and 2 , i.e. it consists of all the real numbers between 1 and 2. But the bounding numbers of an interval need not be integers. An interval of values of the argument contains a number a , when a is a member of the interval. For example, the interval between 1 and 2 contains 3/2 , 5/3 , 7/4 , and so on.
A set of numbers approximates to a number a
within a standard k , when the numerical difference between a and every number of the set is less than k. Here k is the "standard of approximation." Thus the set of numbers 3 , 4 , 6 , 8 , approximates to the number 5 within the standard 4. In this case the standard 4 is not the smallest which could have been chosen, the set also approximates
to 5 within any of the standards 3.1 or 3.01 or 3.001. Again, the numbers, 3.1 , 3.141 , 3.1415 , 3.14159 approximate to 3.13102 within the standard .032 , and also within the smaller standard .03103.
These two ideas of an interval and of
approximation to a number within a standard are easy enough; their only difficulty is that they look rather trivial. But when combined with the next idea, that of the "neighbourhood" of a number, they form the foundation of modern mathematical reasoning. What do we mean by saying that something is true for a function f(x) in the neighbourhood of the value a of the argument x ? It is this fundamental notion which we have now got to make precise.
The values of a function f(x) are said to possess a characteristic in the "neighbourhood of a " when some interval can be found, which (i) contains the number a not as an end-point, and (ii) is such that every value
of the function for arguments, other than a , lying within that interval possesses the characteristic. The value f(a) of the function for the argument a may or may not possess the characteristic. Nothing is decided on this point by statements about the neighbourhood of a.
For example, suppose we take the particular function x^2. Now in the neighbourhood of 2 , the values of x^2 are less than 5. For we can find an interval, e.g. from 1 to 2.1 , which (i) contains 2 not as an end-point, and (ii) is such that, for values of x lying within it, x^2 is less than 5.
Now, combining the preceding ideas we know what is meant by saying that in the neighbourhood of a the function f(x) approximates to c within the standard k. It means that some interval can be found which (i) includes a not as an end-point, and (ii) is such that all values of f(x) , where x lies in the interval and is not a , differ from c by less than k. For example, in the neighbourhood of 2 , the function sqrt(x) approximates to 1.41425 within the standard .0001. This is true because the square root of 1.99996164 is 1.4142 and the square root of 2.00024449 is 1.4143 ; hence for values of x lying in the interval 1.99996164 to 2.00024449 , which contains 2 not as an end-point, the values of the function sqrt(x) all lie between 1.4142 and 1.4143 , and
they therefore all differ from 1.41425 by less than .0001. In this case we can, if we like, fix a smaller standard of approximation, namely .000051 or .0000501. Again, to take another example, in the neighbourhood of 2 the function x^2 approximates to 4 within the standard .5. For (1.9)^2 = 3.61 and (2.1)^2 = 4.41 , and thus the required interval 1.9 to 2.1 , containing 2 not as an end-point, has been found. This example brings out the fact that statements about a function f(x) in the neighbourhood of a number a are distinct from statements about the value of f(x) when x = a. The production of an interval, throughout which the statement is true, is required. Thus the mere fact that 2^2 = 4 does not by itself justify us in saying that in the neighbourhood of 2 the function x^2 is equal to 4. This statement would be untrue, because no interval can be produced with the required property. Also, the fact that 2^2 = 4 does not by itself justify us in saying that in the neighbourhood of 2 the function x^2 approximates to 4 within the standard .5 ; although as a matter of fact, the statement has just been proved to be true.
If we understand the preceding ideas, we understand the foundations of modern mathematics. We shall recur to analogous ideas in the chapter on Series, and again in the chapter on the Differential Calculus.
Meanwhile, we are now prepared to define "continuous functions." A function f(x) is "continuous" at a value a of its argument, when in the neighbourhood of a its values approximate to f(a) (i.e. to its value at a ) within every standard of approximation.
This means that, whatever standard k be chosen, in the neighbourhood of a f(x) approximates to f(a) within the standard k. For example, x^2 is continuous at the value 2 of its argument, x , because however k be chosen we can always find an interval, which (i) contains 2 not as an end-point, and (ii) is such that the values of x^2 for arguments lying within it approximate to 4 (i.e. 2^2 ) within the standard k. Thus, suppose we choose the standard .1 ; now (1.999)^2 = 3.996001 , and (2.01)^2 = 4.0401 , and both these numbers differ from 4 by less than .1. Hence, within the interval 1.999 to 2.01 the values of x^2 approximate to 4 within the standard .1. Similarly an interval can be produced for any other standard which we like to try.
Take the example of the railway train. Its velocity is continuous as it passes the signal box, if whatever velocity you like to assign (say one-millionth of a mile per hour) an interval of time can be found extending before and after the instant of passing, such that at all instants within it the train's velocity
differs from that with which the train passed the box by less than one-millionth of a mile per hour; and the same is true whatever other velocity be mentioned in the place of one-millionth of a mile per hour.
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End of part 11 of 19
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