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89. Limits as tends to . We shall now return to functions of a continuous real variable. We shall confine ourselves entirely to one-valued functions,51 and we shall denote such a function by . We suppose to assume successively all values corresponding to points on our fundamental straight line , starting from some definite point on the line and progressing always to the right. In these circumstances we say that tends to infinity, or to , and write . The only difference between the ‘tending of to ’ discussed in the last chapter, and this ‘tending of to ’, is that assumes all values as it tends to , i.e. that the point which corresponds to coincides in turn with every point of to the right of its initial position, whereas tended to by a series of jumps. We can express this distinction by saying that tends continuously to .
As we explained at the beginning of the last chapter, there is a very close correspondence between functions of and functions of . Every function of may be regarded as a selection from the values of a function of . In the last chapter we discussed the peculiarities which may characterise the behaviour of a function as tends to . Now we are concerned with the same problem for a function ; and the definitions and theorems to which we are led are practically repetitions of those of the last chapter. Thus corresponding to Def. 1 of § 58 we have:
DEFINITION 1. The function is said to tend to the limit as tends to if, when any positive number , however small, is assigned, a number can be chosen such that, for all values of equal to or greater than , differs from by less than , i.e. if when .
When this is the case we may write
or, when there is no risk of ambiguity, simply , or . Similarly we have:DEFINITION 2. The function is said to tend to with if, when any number , however large, is assigned, we can choose a number such that when .
We then write
Similarly we define .52 Finally we have:DEFINITION 3. If the conditions of neither of the two preceding definitions are satisfied, then is said to oscillate as tends to . If is less than some constant when ,53 then is said to oscillate finitely, and otherwise infinitely.
The reader will remember that in the last chapter we considered very carefully various less formal ways of expressing the facts represented by the formulae , . Similar modes of expression may of course be used in the present case. Thus we may say that is small or nearly equal to or large when is large, using the words ‘small’, ‘nearly’, ‘large’ in a sense similar to that in which they were used in Ch. IV.
Examples XXXIV. 1. Consider the behaviour of the following functions as : , , , , , , .
The first four functions correspond exactly to functions of fully discussed in Ch. IV. The graphs of the last three were constructed in Ch. II (Exs. XVI. 1, 2, 4), and the reader will see at once that , oscillates finitely, and .
One simple remark may be inserted here. The function oscillates between and , as is obvious from the form of its graph. It is equal to zero whenever is an integer, so that the function derived from it is always zero and so tends to the limit zero. The same is true if
It is evident that or or involves the corresponding property for , but that the converse is by no means always true.2. Consider in the same way the functions:
illustrating your remarks by means of the graphs of the functions.3. Give a geometrical explanation of Def. 1, analogous to the geometrical explanation of Ch. IV, §59.
4. If , and is not zero, then and oscillate finitely. If or , then they oscillate infinitely. The graph of either function is a wavy curve oscillating between the curves and .
5. Discuss the behaviour, as , of the function
where and are some pair of simple functions (e.g. and ). [The graph of is a curve oscillating between the curves , .]90. Limits as tends to . The reader will have no difficulty in framing for himself definitions of the meaning of the assertions ‘ tends to ’, or ‘’ and
In fact, if and , then tends to as tends to , and the question of the behaviour of as tends to is the same as that of the behaviour of as tends to .91. Theorems corresponding to those of Ch. IV, §§63–67. The theorems concerning the sums, products, and quotients of functions proved in Ch. IV are all true (with obvious verbal alterations which the reader will have no difficulty in supplying) for functions of the continuous variable . Not only the enunciations but the proofs remain substantially the same.
92. Steadily increasing or decreasing functions. The definition which corresponds to that of §69 is as follows: the function will be said to increase steadily with if whenever . In many cases, of course, this condition is only satisfied from a definite value of onwards, i.e. when . The theorem which follows in that section requires no alteration but that of into : and the proof is the same, except for obvious verbal changes.
If , the possibility of equality being excluded, whenever , then will be said to be steadily increasing in the stricter sense. We shall find that the distinction is often important (cf. §§108–109).
The reader should consider whether or no the following functions increase steadily with (or at any rate increase steadily from a certain value of onwards): , , , , , , . All these functions tend to as .
93. Limits as tends to . Let be such a function of that , and let . Then
say. As tends to , tends to the limit , and tends to the limit .Let us now dismiss and consider simply as a function of . We are for the moment concerned only with those values of which correspond to large positive values of , that is to say with small positive values of . And has the property that by making sufficiently small we can make differ by as little as we please from . To put the matter more precisely, the statement expressed by means that, when any positive number , however small, is assigned, we can choose so that for all values of greater than or equal to . But this is the same thing as saying that we can choose so that for all positive values of less than or equal to .
We are thus led to the following definitions:
A. If, when any positive number , however small, is assigned, we can choose so that
when , then we say that tends to the limit as tends to by positive values, and we writeB. If, when any number , however large, is assigned, we can choose so that
when , then we say that tends to as tends to by positive values, and we writeWe define in a similar way the meaning of ‘ tends to the limit as tends to by negative values’, or ‘ when ’. We have in fact only to alter to in definition A. There is of course a corresponding analogue of definition B, and similar definitions in which
as or .If and , we write simply
This case is so important that it is worth while to give a formal definition.If, when any positive number , however small, is assigned, we can choose so that, for all values of different from zero but numerically less than or equal to , differs from by less than , then we say that tends to the limit as tends to , and write
So also, if as and also as , we say that as . We define in a similar manner the statement that as .
Finally, if does not tend to a limit, or to , or to , as , we say that oscillates as , finitely or infinitely as the case may be; and we define oscillation as in a similar manner.
The preceding definitions have been stated in terms of a variable denoted by : what letter is used is of course immaterial, and we may suppose written instead of throughout them.
94. Limits as tends to . Suppose that as , and write
If then and , and we are naturally led to write or simply or , and to say that tends to the limit as tends to . The meaning of this equation may be formally and directly defined as follows: if, given , we can always determine so that when , thenBy restricting ourselves to values of greater than , i.e. by replacing by , we define ‘ tends to when approaches from the right’, which we may write as
In the same way we can define the meaning of Thus is equivalent to the two assertionsWe can give similar definitions referring to the cases in which or as through values greater or less than ; but it is probably unnecessary to dwell further on these definitions, since they are exactly similar to those stated above in the special case when , and we can always discuss the behaviour of as by putting and supposing that .
95. Steadily increasing or decreasing functions. If there is a number such that whenever , then will be said to increase steadily in the neighbourhood of .
Suppose first that , and put . Then as , and is a steadily increasing function of , never greater than . It follows from §92 that tends to a limit not greater than . We shall write
We can define in a similar manner; and it is clear that It is obvious that similar considerations may be applied to decreasing functions.If , the possibility of equality being excluded, whenever , then will be said to be steadily increasing in the stricter sense.
96. Limits of indetermination and the principle of convergence. All of the argument of §§80–84 may be applied to functions of a continuous variable which tends to a limit . In particular, if is bounded in an interval including (i.e. if we can find , , and so that when ).54 then we can define and , the lower and upper limits of indetermination of as , and prove that the necessary and sufficient condition that as is that . We can also establish the analogue of the principle of convergence, i.e. prove that the necessary and sufficient condition that should tend to a limit as is that, when is given, we can choose so that when .
as , then , , and , unless in the last case .[We saw in §91 that the theorems of Ch. IV, §§63 et seq. hold also for functions of when or . By putting we may extend them to functions of , when , and by putting to functions of , when .
The reader should however try to prove them directly from the formal definition given above. Thus, in order to obtain a strict direct proof of the first result he need only take the proof of Theorem I of §63 and write throughout for , for and for .]
2. If is a positive integer then as .
3. If is a negative integer then as , while or as , according as is odd or even. If then and .
4. .
5. , unless . If and , , then the function tends to or , as , according as and have like or unlike signs; the case is reversed if . The case in which both and vanish is considered in Ex. XXXVI. 5. Discuss the cases which arise when and more than one of the first coefficients in the denominator vanish.
6. , if is any positive or negative integer, except when and is negative. [If , put and apply Ex. 4. When , the result follows from Ex. 1 above. It follows at once that , if is any polynomial.]
7. , if denotes any rational function and is not one of the roots of its denominator.
8. Show that for all rational values of , except when and is negative. [This follows at once, when is positive, from the inequalities (9) or (10) of §74. For , where is the greater of the absolute values of and (cf. Ex. XXVIII. 4). If is negative we write and . Then
97. The reader will probably fail to see at first that any proof of such results as those of Exs. 4, 5, 6, 7, 8 above is necessary. He may ask ‘why not simply put , or ? Of course we then get , , , , ’. It is very important that he should see exactly where he is wrong. We shall therefore consider this point carefully before passing on to any further examples.
The statement
is a statement about the values of when has any value distinct from but differing by little from zero.55 It is not a statement about the value of when . When we make the statement we assert that, when is nearly equal to zero, is nearly equal to . We assert nothing whatever about what happens when is actually equal to . So far as we know, may not be defined at all for ; or it may have some value other than . For example, consider the function defined for all values of by the equation . It is obvious that(1) |
Now consider the function which differs from only in that when . Then
(2) |
for, when is nearly equal to zero, is not only nearly but exactly equal to zero. But . The graph of this function consists of the axis of , with the point left out, and one isolated point, viz. the point . The equation (2) expresses the fact that if we move along the graph towards the axis of , from either side, then the ordinate of the curve, being always equal to zero, tends to the limit zero. This fact is in no way affected by the position of the isolated point .
The reader may object to this example on the score of artificiality: but it is easy to write down simple formulae representing functions which behave precisely like this near . One is
where denotes as usual the greatest integer not greater than . For if then ; while if , or , then and so .Or again, let us consider the function
already discussed in Ch. II, § 24, (2). This function is equal to for all values of save . It is not equal to when : it is in fact not defined at all for . For when we say that is defined for we mean (as we explained in Ch. II, l.c.) that we can calculate its value for by putting in the actual expression of . In this case we cannot. When we put in we obtain , which is a meaningless expression. The reader may object ‘divide numerator and denominator by ’. But he must admit that when this is impossible. Thus is a function which differs from solely in that it is not defined for . None the less for is equal to so long as differs from zero, however small the difference may be.Similarly so long as is not equal to zero, but is undefined when . None the less .
On the other hand there is of course nothing to prevent the limit of as tends to zero from being equal to , the value of for . Thus if then and . This is in fact, from a practical point of view, i.e. from the point of view of what most frequently occurs in applications, the ordinary case.
2. , if is any integer (zero included).
3. Show that the result of Ex. 2 remains true for all rational values of , provided is positive. [This follows at once from the inequalities (9) and (10) of §74.]
4. . [Observe that is a factor of both numerator and denominator.]
5. Discuss the behaviour of
as tends to by positive or negative values.[If , . If , . If and is even, or according as or . If and is odd, as and as , or as and as , according as or .]
6. Orders of smallness. When is small is very much smaller, much smaller still, and so on: in other words
Another way of stating the matter is to say that, when tends to , , , … all also tend to , but tends to more rapidly than , than , and so on. It is convenient to have some scale by which to measure the rapidity with which a function, whose limit, as tends to , is , diminishes with , and it is natural to take the simple functions , , , … as the measures of our scale.
We say, therefore, that is of the first order of smallness if tends to a limit other than as tends to . Thus is of the first order of smallness, since .
Similarly we define the second, third, fourth, … orders of smallness. It must not be imagined that this scale of orders of smallness is in any way complete. If it were complete, then every function which tends to zero with would be of either the first or second or some higher order of smallness. This is obviously not the case. For example tends to zero more rapidly than and less rapidly than .
The reader may not unnaturally think that our scale might be made complete by including in it fractional orders of smallness. Thus we might say that was of the th order of smallness. We shall however see later on that such a scale of orders would still be altogether incomplete. And as a matter of fact the integral orders of smallness defined above are so much more important in applications than any others that it is hardly necessary to attempt to make our definitions more precise.
Orders of greatness. Similar definitions are at once suggested to meet the case in which is large (positively or negatively) when is small. We shall say that is of the th order of greatness when is small if tends to a limit different from as tends to .
These definitions have reference to the case in which . There are of course corresponding definitions relating to the cases in which or . Thus if tends to a limit other than zero, as , then we say that is of the th order of smallness when is large: while if tends to a limit other than zero, as , then we say that is of the th order of greatness when is nearly equal to .
7.56. [Put or , and use Ex. XXXV. 8.]
8. . [Multiply numerator and denominator by .]
9. Consider the behaviour of as , and being positive integers.
10. .
11. .
12. Draw a graph of the function
Has it a limit as ? [Here except for , , , , when is not defined, and as .]
13. .
[It may be deduced from the definitions of the trigonometrical ratios57 that if is positive and less than then
or orBut . Hence , and . As is an even function, the result follows.]
14. .
15. . Is this true if ?
16. . [Put .]
17. , .
18. .
20. How do the functions , , behave as ? [The first oscillates finitely, the second infinitely, the third tends to the limit . None is defined when . See Exs. XV. 6, 7, 8.]
21. Does the function
tend to a limit as tends to ? [No. The function is equal to except when ; i.e. when , , …, , , …. For these values the formula for assumes the meaningless form , and is therefore not defined for an infinity of values of near .]22. Prove that if is any integer then and as , and , as .
98. Continuous functions of a real variable. The reader has no doubt some idea as to what is meant by a continuous curve. Thus he would call the curve in Fig. 29 continuous, the curve generally continuous but discontinuous for and .
Either of these curves may be regarded as the graph of a function . It is natural to call a function continuous if its graph is a continuous curve, and otherwise discontinuous. Let us take this as a provisional definition and try to distinguish more precisely some of the properties which are involved in it.
In the first place it is evident that the property of the function of which is the graph may be analysed into some property possessed by the curve at each of its points. To be able to define continuity for all values of we must first define continuity for any particular value of . Let us therefore fix on some particular value of , say the value corresponding to the point of the graph. What are the characteristic properties of associated with this value of ?
In the first place is defined for . This is obviously essential. If were not defined there would be a point missing from the curve.
Secondly is defined for all values of near ; i.e. we can find an interval, including in its interior, for all points of which is defined.
Thirdly if approaches the value from either side then approaches the limit .
The properties thus defined are far from exhausting those which are possessed by the curve as pictured by the eye of common sense. This picture of a curve is a generalisation from particular curves such as straight lines and circles. But they are the simplest and most fundamental properties: and the graph of any function which has these properties would, so far as drawing it is practically possible, satisfy our geometrical feeling of what a continuous curve should be. We therefore select these properties as embodying the mathematical notion of continuity. We are thus led to the following
DEFINITION. The function is said to be continuous for if it tends to a limit as tends to from either side, and each of these limits is equal to .
We can now define continuity throughout an interval. The function is said to be continuous throughout a certain interval of values of if it is continuous for all values of in that interval. It is said to be continuous everywhere if it is continuous for every value of . Thus is continuous in the interval , where is any positive number less than ; and and are continuous everywhere.
If we recur to the definitions of a limit we see that our definition is equivalent to ‘ is continuous for if, given , we can choose so that if ’.
We have often to consider functions defined only in an interval . In this case it is convenient to make a slight and obvious change in our definition of continuity in so far as it concerns the particular points and . We shall then say that is continuous for if exists and is equal to , and for if exists and is equal to .
99. The definition of continuity given in the last section may be illustrated geometrically as follows. Draw the two horizontal lines and . Then expresses the fact that the point on the curve corresponding to lies
between these two lines. Similarly expresses the fact that lies in the interval . Thus our definition asserts that if we draw two such horizontal lines, no matter how close together, we can always cut off a vertical strip of the plane by two vertical lines in such a way that all that part of the curve which is contained in the strip lies between the two horizontal lines. This is evidently true of the curve (Fig. 29), whatever value may have.
We shall now discuss the continuity of some special types of functions. Some of the results which follow were (as we pointed out at the time) tacitly assumed in Ch. II.
Examples XXXVII. 1. The sum or product of two functions continuous at a point is continuous at that point. The quotient is also continuous unless the denominator vanishes at the point. [This follows at once from Ex. XXXV. 1.]
2. Any polynomial is continuous for all values of . Any rational fraction is continuous except for values of for which the denominator vanishes. [This follows from Exs. XXXV. 6, 7.]
3. is continuous for all positive values of (Ex. XXXV. 8). It is not defined when , but is continuous for in virtue of the remark made at the end of §98. The same is true of , where and are any positive integers of which is even.
4. The function , where is odd, is continuous for all values of .
5. is not continuous for . It has no value for , nor does it tend to a limit as . In fact or according as by positive or negative values.
6. Discuss the continuity of , where and are positive integers, for .
7. The standard rational function is discontinuous for , where is any root of . Thus is discontinuous for . It will be noticed that in the case of rational functions a discontinuity is always associated with (a) a failure of the definition for a particular value of and (b) a tending of the function to or as approaches this value from either side. Such a particular kind of point of discontinuity is usually described as an infinity of the function. An ‘infinity’ is the kind of discontinuity of most common occurrence in ordinary work.
8. Discuss the continuity of
9. and are continuous for all values of .
[We have
which is numerically less than the numerical value of .]10. For what values of are , , , and continuous or discontinuous?
11. If is continuous for , and is a continuous function of which is equal to when , then is continuous for .
12. If is continuous for any particular value of , then any polynomial in , such as , is so too.
13. Discuss the continuity of
14. , , and are continuous except for .
15. The function which is equal to except when , and to zero when , is continuous for all values of .
16. and are discontinuous for all integral values of .
17. For what (if any) values of are the following functions discontinuous: , , , , , ?
18. Classification of discontinuities. Some of the preceding examples suggest a classification of different types of discontinuity.
(1) Suppose that tends to a limit as either by values less than or by values greater than . Denote these limits, as in §95, by and respectively. Then, for continuity, it is necessary and sufficient that should be defined for , and that . Discontinuity may arise in a variety of ways.
() may be equal to , but may not be defined, or may differ from and . Thus if and , , but is not defined for . Or if and , , but .
() and may be unequal. In this case may be equal to one or to neither, or be undefined. The first case is illustrated by , for which , ; the second by , for which , , ; and the third by , for which , , and is undefined.
In any of these cases we say that has a simple discontinuity at . And to these cases we may add those in which is defined only on one side of , and or , as the case may be, exists, but is either not defined when or has when a value different from or .
It is plain from §95 that a function which increases or decreases steadily in the neighbourhood of can have at most a simple discontinuity for .
(2) It may be the case that only one (or neither) of and exists, but that, supposing for example not to exist, or as , so that tends to a limit or to or to as approaches from either side. Such is the case, for instance, if or , and . In such cases we say (cf. Ex. 7) that is an infinity of . And again we may add to these cases those in which or as from one side, but is not defined at all on the other side of .
(3) Any point of discontinuity which is not a point of simple discontinuity nor an infinity is called a point of oscillatory discontinuity. Such is the point for the functions , .
19. What is the nature of the discontinuities at of the functions , , , , , , ?
20. The function which is equal to when is rational and to when is irrational (Ch. II, Ex. XVI. 10) is discontinuous for all values of . So too is any function which is defined only for rational or for irrational values of .
21. Thefunctionwhichisequalto when is irrational and to when is a rational fraction (Ch. II, Ex. XVI. 11) is discontinuous for all negative and for positive rational values of , but continuous for positive irrational values.
22. For what points are the functions considered in Ch. IV, Exs. XXXI discontinuous, and what is the nature of their discontinuities? [Consider, e.g., the function (Ex. 5). Here is only defined when : it is equal to when and to when . The points and are points of simple discontinuity.]
100. The fundamental property of a continuous function. It may perhaps be thought that the analysis of the idea of a continuous curve given in § 98 is not the simplest or most natural possible. Another method of analysing our idea of continuity is the following. Let and be two points on the graph of whose coordinates are , and , respectively. Draw any straight line which passes between and . Then common sense certainly declares that if the graph of is continuous it must cut .
If we consider this property as an intrinsic geometrical property of continuous curves it is clear that there is no real loss of generality in supposing to be parallel to the axis of . In this case the ordinates of and cannot be equal: let us suppose, for definiteness, that . And let be the line , where . Then to say that the graph of must cut is the same thing as to say that there is a value of between and for which .
We conclude then that a continuous function must possess the following property: if and , then there is a value of between and for which . In other words as varies from to , must assume at least once every value between and .
We shall now prove that if is a continuous function of in the sense defined in § 98 then it does in fact possess this property. There is a certain range of values of , to the right of , for which . For , and so is certainly less than if is numerically less than . But since is continuous for , this condition is certainly satisfied if is near enough to . Similarly there is a certain range of values, to the left of , for which .
Let us divide the values of between and into two classes , as follows:
(1) in the class we put all values of such that when and for all values of between and ;
(2) in the class we put all the other values of , i.e. all numbers such that either or there is a value of between and for which .
Then it is evident that these two classes satisfy all the conditions imposed upon the classes , of § 17, and so constitute a section of the real numbers. Let be the number corresponding to the section.
First suppose , so that belongs to the upper class: and let , say. Then and so
for all values of less than , which contradicts the condition of continuity for .Next suppose . Then, if is any number greater than , either or we can find a number between and such that . In either case we can find a number as near to as we please and such that the corresponding values of differ by more than . And this again contradicts the hypothesis that is continuous for .
Hence , and the theorem is established. It should be observed that we have proved more than is asserted explicitly in the theorem; we have proved in fact that is the least value of for which . It is not obvious, or indeed generally true, that there is a least among the values of for which a function assumes a given value, though this is true for continuous functions.
It is easy to see that the converse of the theorem just proved is not true. Thus such a function as the function whose graph is represented by Fig. 31 obviously assumes at least once every value between and : yet is discontinuous. Indeed it is not even true that must be continuous when it assumes each value once and once only. Thus let be defined as follows from to . If let ; if let ; and if let . The graph of the function is shown in Fig. 32; it includes the points , but not the points , . It is clear that, as varies from to , assumes once and once only every value between and ; but is discontinuous for and .
As a matter of fact, however, the curves which usually occur in elementary mathematics are composed of a finite number of pieces along which always varies in the same direction. It is easy to show that if always varies in the same direction, i.e. steadily increases or decreases, as varies from to , then the two notions of continuity are really equivalent, i.e. that if takes every value between and then it must be a continuous function in the sense of §98. For let be any value of between and . As through values less than , tends to the limit (§95). Similarly as through values greater than , tends to the limit . The function will be continuous for if and only if
But if either of these equations is untrue, say the first, then it is evident that never assumes any value which lies between and , which is contrary to our assumption. Thus must be continuous. The net result of this and the last section is consequently to show that our common-sense notion of what we mean by continuity is substantially accurate, and capable of precise statement in mathematical terms.101. In this and the following paragraphs we shall state and prove some general theorems concerning continuous functions.
THEOREM 1. Suppose that is continuous for , and that is positive. Then we can determine a positive number such that is positive throughout the interval .
For, taking in the fundamental inequality of p. 572, we can choose so that
throughout , and then so that is positive. There is plainly a corresponding theorem referring to negative values of .THEOREM 2. If is continuous for , and vanishes for values of as near to as we please, or assumes, for values of as near to as we please, both positive and negative values, then .
This is an obvious corollary of Theorem 1. If is not zero, it must be positive or negative; and if it were, for example, positive, it would be positive for all values of sufficiently near to , which contradicts the hypotheses of the theorem.
102. The range of values of a continuous function. Let us consider a function about which we shall only assume at present that it is defined for every value of in an interval .
The values assumed by for values of in form an aggregate to which we can apply the arguments of § 80, as we applied them in § 81 to the aggregate of values of a function of . If there is a number such that , for all values of in question, we say that is bounded above. In this case possesses an upper bound : no value of exceeds , but any number less than is exceeded by at least one value of . Similarly we define ‘bounded below’, ‘lower bound’, ‘bounded’, as applied to functions of a continuous variable .
THEOREM 1. If is continuous throughout , then it is bounded in .
We can certainly determine an interval , extending to the right from , in which is bounded. For since is continuous for , we can, given any positive number however small, determine an interval throughout which lies between and ; and obviously is bounded in this interval.
Now divide the points of the interval into two classes , , putting in if is bounded in , and in if this is not the case. It follows from what precedes that certainly exists: what we propose to prove is that does not. Suppose that does exist, and let be the number corresponding to the section whose lower and upper classes are and . Since is continuous for , we can, however small may be, determine an interval 58 throughout which
Thus is bounded in . Now belongs to . Therefore is bounded in : and therefore it is bounded in the whole interval . But belongs to and so is not bounded in . This contradiction shows that does not exist. And so is bounded in the whole interval .THEOREM 2. If is continuous throughout , and and are its upper and lower bounds, then assumes the values and at least once each in the interval.
For, given any positive number , we can find a value of for which or . Hence is not bounded, and therefore, by Theorem 1, is not continuous. But is a continuous function, and so is continuous at any point at which its denominator does not vanish (Ex. XXXVII. 1). There must therefore be one point at which the denominator vanishes: at this point . Similarly it may be shown that there is a point at which .
The proof just given is somewhat subtle and indirect, and it may be well, in view of the great importance of the theorem, to indicate alternative lines of proof. It will however be convenient to postpone these for a moment.59
Examples XXXVIII. 1. If except when , and when , then has neither an upper nor a lower bound in any interval which includes in its interior, as e.g. the interval .
2. If except when , and when , then has the lower bound , but no upper bound, in the interval .
3. Let except when , and when . Then is discontinuous for . In any interval the lower bound is and the upper bound , and each of these values is assumed by an infinity of times.
4. Let . This function is discontinuous for all integral values of . In the interval its lower bound is and its upper bound . It is equal to when or , but it is never equal to . Thus never assumes a value equal to its upper bound.
5. Let when is irrational, and when is a rational fraction . Then has the lower bound , but no upper bound, in any interval . But if when , then has neither an upper nor a lower bound in any interval.
103. The oscillation of a function in an interval. Let be any function bounded throughout , and and its upper and lower bounds. We shall now use the notation , for , , in order to exhibit explicitly the dependence of and on and , and we shall write
This number , the difference between the upper and lower bounds of in , we shall call the oscillation of in . The simplest of the properties of the functions , , are as follows.
(1) If then is equal to the greater of and , and to the lesser of and .
(2) is an increasing, a decreasing, and an increasing function of .
(3) .
The first two theorems are almost immediate consequences of our definitions. Let be the greater of and , and let be any positive number. Then throughout and , and therefore throughout ; and somewhere in or in , and therefore somewhere in . Hence . The proposition concerning may be proved similarly. Thus (1) is proved, and (2) is an obvious corollary.
Suppose now that is the greater and the less of and , and that is the less and the greater of and . Then, since belongs to both intervals, is not greater than nor less than . Hence , whether these numbers correspond to the same one of the intervals and or not, and
But and (3) follows.104. Alternative proofs of Theorem 2 of §102. The most straightforward proof of Theorem 2 of §102 is as follows. Let be any number of the interval . The function increases steadily with and never exceeds . We can therefore construct a section of the numbers by putting in or in according as or . Let be the number corresponding to the section. If , we have
for all positive values of , and so by (1) of §103. Hence assumes, for values of as near as we please to , values as near as we please to , and so, since is continuous, must be equal to .If then . And if then , and so . In either case the argument may be completed as before.
The theorem may also be proved by the method of repeated bisection used in §71. If is the upper bound of in an interval , and is divided into two equal parts, then it is possible to find a half in which the upper bound of is also . Proceeding as in §71, we construct a sequence of intervals , , , … in each of which the upper bound of is . These intervals, as in §71, converge to a point , and it is easily proved that the value of at this point is .
105. Sets of intervals on a line. The Heine-Borel Theorem. We shall now proceed to prove some theorems concerning the oscillation of a function which are of a somewhat abstract character but of very great importance, particularly, as we shall see later, in the theory of integration. These theorems depend upon a general theorem concerning intervals on a line.
Suppose that we are given a set of intervals in a straight line, that is to say an aggregate each of whose members is an interval . We make no restriction as to the nature of these intervals; they may be finite or infinite in number; they may or may not overlap;60 and any number of them may be included in others.
It is worth while in passing to give a few examples of sets of intervals to which we shall have occasion to return later.
(i) If the interval is divided into equal parts then the intervals thus formed define a finite set of non-overlapping intervals which just cover up the line.
(ii) We take every point of the interval , and associate with the interval , where is a positive number less than , except that with we associate and with we associate , and in general we reject any part of any interval which projects outside the interval . We thus define an infinite set of intervals, and it is obvious that many of them overlap with one another.
(iii) We take the rational points of the interval , and associate with the interval
where is positive and less than . We regard as and as : in these two cases we reject the part of the interval which lies outside . We obtain thus an infinite set of intervals, which plainly overlap with one another, since there are an infinity of rational points, other than , in the interval associated with .The Heine-Borel Theorem. Suppose that we are given an interval , and a set of intervals each of whose members is included in . Suppose further that possesses the following properties:
(i) every point of , other than and , lies inside61 at least one interval of ;
(ii) is the left-hand end point, and the right-hand end point, of at least one interval of .
Then it is possible to choose a finite number of intervals from the set which form a set of intervals possessing the properties (i) and (ii).
We know that is the left-hand end point of at least one interval of , say . We know also that lies inside at least one interval of , say . Similarly lies inside an interval of . It is plain that this argument may be repeated indefinitely, unless after a finite number of steps coincides with .
If does coincide with after a finite number of steps then there is nothing further to prove, for we have obtained a finite set of intervals, selected from the intervals of , and possessing the properties required. If never coincides with , then the points , , , … must (since each lies to the right of its predecessor) tend to a limiting position, but this limiting position may, so far as we can tell, lie anywhere in .
Let us suppose now that the process just indicated, starting from , is performed in all possible ways, so that we obtain all possible sequences of the type , , , …. Then we can prove that there must be at least one such sequence which arrives at after a finite number of steps.
There are two possibilities with regard to any point between and . Either (i) lies to the left of some point of some sequence or (ii) it does not. We divide the points into two classes and according as to whether (i) or (ii) is true. The class certainly exists, since all points of the interval belong to . We shall now prove that does not exist, so that every point belongs to .
If exists then lies entirely to the left of , and the classes , form a section of the real numbers between and , to which corresponds a number . The point lies inside an interval of , say , and belongs to , and so lies to the left of some term of some sequence. But then we can take as the interval associated with in our construction of the sequence , , , …; and all points to the left of lie to the left of . There are therefore points of to the right of , and this contradicts the definition of . It is therefore impossible that should exist.
Thus every point belongs to . Now is the right-hand end point of an interval of , say , and belongs to . Hence there is a member of a sequence , , , … such that . But then we may take the interval corresponding to to be , and so we obtain a sequence in which the term after the th coincides with , and therefore a finite set of intervals having the properties required. Thus the theorem is proved.
It is instructive to consider the examples of p. 603 in the light of this theorem.
(i) Here the conditions of the theorem are not satisfied: the points , , , … do not lie inside any interval of .
(ii) Here the conditions of the theorem are satisfied. The set of intervals
associated with the points , , , …, , possesses the properties required.(iii) In this case we can prove, by using the theorem, that there are, if is small enough, points of which do not lie in any interval of .
If every point of lay inside an interval of (with the obvious reservation as to the end points), then we could find a finite number of intervals of possessing the same property and having therefore a total length greater than . Now there are two intervals, of total length , for which , and intervals, of total length , associated with any other value of . The sum of any finite number of intervals of can therefore not be greater than times that of the series
which will be shown to be convergent in Ch. VIII. Hence it follows that, if is small enough, the supposition that every point of lies inside an interval of leads to a contradiction.The reader may be tempted to think that this proof is needlessly elaborate, and that the existence of points of the interval, not in any interval of , follows at once from the fact that the sum of all these intervals is less than . But the theorem to which he would be appealing is (when the set of intervals is infinite) far from obvious, and can only be proved rigorously by some such use of the Heine-Borel Theorem as is made in the text.
106. We shall now apply the Heine-Borel Theorem to the proof of two important theorems concerning the oscillation of a continuous function.
THEOREM I. If is continuous throughout the interval , then we can divide into a finite number of sub-intervals , , …, , in each of which the oscillation of is less than an assigned positive number .
Let be any number between and . Since is continuous for , we can determine an interval such that the oscillation of in this interval is less than . It is indeed obvious that there are an infinity of such intervals corresponding to every and every , for if the condition is satisfied for any particular value of , then it is satisfied a fortiori for any smaller value. What values of are admissible will naturally depend upon ; we have at present no reason for supposing that a value of admissible for one value of will be admissible for another. We shall call the intervals thus associated with the -intervals of .
If then we can determine an interval , and so an infinity of such intervals, having the same property. These we call the -intervals of , and we can define in a similar manner the -intervals of .
Consider now the set of intervals formed by taking all the -intervals of all points of . It is plain that this set satisfies the conditions of the Heine-Borel Theorem; every point interior to the interval is interior to at least one interval of , and and are end points of at least one such interval. We can therefore determine a set which is formed by a finite number of intervals of , and which possesses the same property as itself.
The intervals which compose the set will in general overlap as in Fig. 34. But their end
points obviously divide up into a finite set of intervals each of which is included in an interval of , and in each of which the oscillation of is less than . Thus Theorem I is proved.
THEOREM II. Given any positive number , we can find a number such that, if the interval is divided in any manner into sub-intervals of length less than , then the oscillation of in each of them will be less than .
Take , and construct, as in Theorem I, a finite set of sub-intervals in each of which the oscillation of is less than . Let be the length of the least of these sub-intervals . If now we divide into parts each of length less than , then any such part must lie entirely within at most two successive sub-intervals . Hence, in virtue of (3) of § 103, the oscillation of , in one of the parts of length less than , cannot exceed twice the greatest oscillation of in a sub-interval , and is therefore less than , and therefore than .
This theorem is of fundamental importance in the theory of definite integrals (Ch. VII). It is impossible, without the use of this or some similar theorem, to prove that a function continuous throughout an interval necessarily possesses an integral over that interval.
107. Continuous functions of several variables. The notions of continuity and discontinuity may be extended to functions of several independent variables (Ch. II, §§ 31 et seq.). Their application to such functions however, raises questions much more complicated and difficult than those which we have considered in this chapter. It would be impossible for us to discuss these questions in any detail here; but we shall, in the sequel, require to know what is meant by a continuous function of two variables, and we accordingly give the following definition. It is a straightforward generalisation of the last form of the definition of § 98.
The function of the two variables and is said to be continuous for , if, given any positive number , however small, we can choose so that
when and ; that is to say if we can draw a square, whose sides are parallel to the axes of coordinates and of length , whose centre is the point , and which is such that the value of at any point inside it or on its boundary differs from by less than .62This definition of course presupposes that is defined at all points of the square in question, and in particular at the point . Another method of stating the definition is this: is continuous for , if when , in any manner. This statement is apparently simpler; but it contains phrases the precise meaning of which has not yet been explained and can only be explained by the help of inequalities like those which occur in our original statement.
It is easy to prove that the sums, the products, and in general the quotients of continuous functions of two variables are themselves continuous. A polynomial in two variables is continuous for all values of the variables; and the ordinary functions of and which occur in every-day analysis are generally continuous, i.e. are continuous except for pairs of values of and connected by special relations.
The reader should observe carefully that to assert the continuity of with respect to the two variables and is to assert much more than its continuity with respect to each variable considered separately. It is plain that if is continuous with respect to and then it is certainly continuous with respect to (or ) when any fixed value is assigned to (or ). But the converse is by no means true. Suppose, for example, that
when neither nor is zero, and when either or is zero. Then if has any fixed value, zero or not, is a continuous function of , and in particular continuous for ; for its value when is zero, and it tends to the limit zero as . In the same way it may be shown that is a continuous function of . But is not a continuous function of and for , . Its value when , is zero; but if and tend to zero along the straight line , then which may have any value between and .108. Implicit functions. We have already, in Ch. II, met with the idea of an implicit function. Thus, if and are connected by the relation
(1) |
then is an ‘implicit function’ of .
But it is far from obvious that such an equation as this does really define a function of , or several such functions. In Ch. II we were content to take this for granted. We are now in a position to consider whether the assumption we made then was justified.
We shall find the following terminology useful. Suppose that it is possible to surround a point , as in §107, with a square throughout which a certain condition is satisfied. We shall call such a square a neighbourhood of , and say that the condition in question is satisfied in the neighbourhood of , or near , meaning by this simply that it is possible to find some square throughout which the condition is satisfied. It is obvious that similar language may be used when we are dealing with a single variable, the square being replaced by an interval on a line.
THEOREM. If (i) is a continuous function of and in the neighbourhood of ,
(ii) ,
(iii) is, for all values of in the neighbourhood of , a steadily increasing function of , in the stricter sense of §95,
then (1) there is a unique function which, when substituted in the equation , satisfies it identically for all values of in the neighbourhood of ,
(2) is continuous for all values of in the neighbourhood of .
In the figure the square represents a ‘neighbourhood’ of throughout which the conditions (i) and (iii) are satisfied, and the point . If we
take and as in the figure, it follows from (iii) that is positive at and negative at . This being so, and being continuous at and at , we can draw lines and parallel to , so that is parallel to and is positive at all points of and negative at all points of . In particular is positive at and negative at , and therefore, in virtue of (iii) and §100, vanishes once and only once at a point on . The same construction gives us a unique point at which on each ordinate between and . It is obvious, moreover, that the same construction can be carried out to the left of . The aggregate of points such as gives us the graph of the required function .
It remains to prove that is continuous. This is most simply effected by using the idea of the ‘limits of indetermination’ of as (§96). Suppose that , and let and be the limits of indetermination of as . It is evident that the points and lie on . Moreover, we can find a sequence of values of such that when through the values of the sequence; and since , and is a continuous function of and , we have
Hence ; and similarly . Thus tends to the limit as , and so is continuous for . It is evident that we can show in exactly the same way that is continuous for any value of in the neighbourhood of .It is clear that the truth of the theorem would not be affected if we were to change ‘increasing’ to ‘decreasing’ in condition (iii).
As an example, let us consider the equation (1), taking , . It is evident that the conditions (i) and (ii) are satisfied. Moreover
has, when , , and are sufficiently small, the sign opposite to that of . Hence condition (iii) (with ‘decreasing’ for ‘increasing’) is satisfied. It follows that there is one and only one continuous function which satisfies the equation (1) identically and vanishes with .The same conclusion would follow if the equation were
The function in question is in this case where the square root is positive. The second root, in which the sign of the square root is changed, does not satisfy the condition of vanishing with .There is one point in the proof which the reader should be careful to observe. We supposed that the hypotheses of the theorem were satisfied ‘in the neighbourhood of ’, that is to say throughout a certain square , . The conclusion holds ‘in the neighbourhood of ’, that is to say throughout a certain interval . There is nothing to show that the of the conclusion is the of the hypotheses, and indeed this is generally untrue.
109. Inverse Functions. Suppose in particular that is of the form . We then obtain the following theorem.
If is a function of , continuous and steadily increasing , in the stricter sense of §95, in the neighbourhood of , and , then there is a unique continuous function which is equal to when and satisfies the equation identically in the neighbourhood of .
The function thus defined is called the inverse function of .
Suppose for example that , , . Then all the conditions of the theorem are satisfied. The inverse function is .
If we had supposed that then the conditions of the theorem would not have been satisfied, for is not a steadily increasing function of in any interval which includes : it decreases when is negative and increases when is positive. And in this case the conclusion of the theorem does not hold, for defines two functions of , viz. and , both of which vanish when , and each of which is defined only for positive values of , so that the equation has sometimes two solutions and sometimes none. The reader should consider the more general equations
in the same way. Another interesting example is given by the equation already considered in Ex. XIV. 7.Similarly the equation
has just one solution which vanishes with , viz. the value of which vanishes with . There are of course an infinity of solutions, given by the other values of (cf. Ex. XV. 10), which do not satisfy this condition.So far we have considered only what happens in the neighbourhood of a particular value of . Let us suppose now that is positive and steadily increasing (or decreasing) throughout an interval . Given any point of , we can determine an interval including , and a unique and continuous inverse function defined throughout .
From the set of intervals we can, in virtue of the Heine-Borel Theorem, pick out a finite sub-set covering up the whole interval ; and it is plain that the finite set of functions , corresponding to the sub-set of intervals thus selected, define together a unique inverse function continuous throughout .
We thus obtain the theorem: if , where is continuous and increases steadily and strictly from to as increases from to , then there is a unique inverse function which is continuous and increases steadily and strictly from to as increases from to .
It is worth while to show how this theorem can be obtained directly without the help of the more difficult theorem of §108. Suppose that , and consider the class of values of such that (i) and (ii) . This class has an upper bound , and plainly . If were less than , we could find a value of such that and , and would not be the upper bound of the class considered. Hence . The equation has therefore a unique solution , say; and plainly increases steadily and continuously with , which proves the theorem.
1. Show that, if neither nor is zero, then
where is of the first order of smallness when is large.2. If , and is not zero, then as increases has ultimately the sign of ; and so has , where is any constant.
3. Show that in general
where , , and is of the first order of smallness when is large. Indicate any exceptional cases.4. Express
in the form where is of the first order of smallness when is large.5. Show that
[Use the formula .]
6. Show that , where is of the first order of smallness when is large.
7. Find values of and such that has the limit zero as ; and prove that .
8. Evaluate
9. Prove that as .
10. Prove that is of the fourth order of smallness when is small; and find the limit of as .
11. Prove that is of the sixth order of smallness when is small; and find the limit of as .
12. From a point on a radius of a circle, produced beyond the circle, a tangent is drawn to the circle, touching it in , and is drawn perpendicular to . Show that as moves up to .
13. Tangents are drawn to a circular arc at its middle point and its extremities; is the area of the triangle formed by the chord of the arc and the two tangents at the extremities, and the area of that formed by the three tangents. Show that as the length of the arc tends to zero.
14. For what values of does tend to (1) , (2) , as ? [To if , to if : the function oscillates if .]
15. If when , and when is irrational, then is continuous for all irrational and discontinuous for all rational values of .
16. Show that the function whose graph is drawn in Fig. 32 may be represented by either of the formulae
17. Show that the function which is equal to when , to when , to when , to when , and to when , assumes every value between and once and once only as increases from to , but is discontinuous for , , and . Show also that the function may be represented by the formula
18. Let when is rational and when is irrational. Show that assumes every value between and once and once only as increases from to , but is discontinuous for every value of except .
19. As increases from to , is continuous and steadily increases, in the stricter sense, from to . Deduce the existence of a function which is a continuous and steadily increasing function of from to .
20. Show that the numerically least value of is continuous for all values of and increases steadily from to as varies through all real values.
21. Discuss, on the lines of §§108–109, the solution of the equations
in the neighbourhood of , .22. If and , then one value of is given by , where
and is of the first order of smallness when is small.[If then
say. It is evident that is of the second order of smallness, of the third, and of the fourth; and , the error being of the fourth order.]23. If then one value of is given by
where , , , and is of the first order of smallness when is small.24. If , where is an integer greater than unity, then one value of is given by , where , , , and is of the th order of smallness when is small.
25. Show that the least positive root of the equation is a continuous function of throughout the interval , and decreases steadily from to as increases from to . [The function is the inverse of : apply §109.]
26. The least positive root of is a continuous function of throughout the interval , and increases steadily from to as increases from towards .
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