Appendix I to Chapter I

Preliminary Remarks: In Greek mathematics, we find an extensive development of the principle that all theorems are to be proved in a logically coherent way by reducing them to a system of axioms, as few in number as possible and not themselves to be proved. At the beginning of the modem era, this axiomatic method of presentation, which at the same time served as a test of the accuracy of the investigation, was regarded as a model for other branches of knowledge. For example, in philosophy, men like Descartes and Spinoza believed that they had made their investigations more convincing by presenting them axiomatically, or, as they called it, geometrically.

But it was a different matter with modern mathematics, which began to develop at about the same time as the new philosophy. In mathematics, the principle of reduction of material to axioms was frequently abandoned. Intuitive evidence in each separate case became a favourite method of proof. Even in the case of scientists of the first rank, we find operations with the new concepts based chiefly on a feeling for the right result and not always free from mystical associations—particularly in the case of the ominous infinitely small quantities or infinitesimals. Blind faith in the omnipotence of the new methods carried the investigator away along paths which he could never have travelled if subjected to the limitations of complete rigour. It is no wonder that only the sure instinct of a great master could guard against gross errors.

It is fortunate that this was so and that the critical counter-currents, which arose in the Eighteenth Century and reached their full strength in the Nineteenth Century, did not come in time to check the development of modem mathematics, but only in time to establish and extend its results. But the need for critical investigation and consolidation of the advances made gradually increased to such an extent that its satisfaction is rightly regarded as one of the most important mathematical achievements of the Nineteenth Century.

In the differential and integral calculus, the critical work of Cauchy is particularly important. By formulating the fundamental concepts in a clear and satisfactory way, Cauchy rounded off in many directions the work, which began in the Eighteenth Century, of presenting higher analysis in an intelligible manner free from the vagueness due to the use of infinitesimals.

The principal thing, which remained to be done, was to replace intuitive considerations in proofs and discussions by considerations of pure analysis, depending only on numbers and on the operations, which can be performed with numbers - as we say, to arithmetize analysis. As a matter of fact, the critically trained mind feels there is something unsatisfactory about appeals to intuition in proofs in analysis. We need not go into the question of the accuracy or inaccuracy of intuition or of the existence of a pure a priori intuition in Kant's sense, in order to recognize that naive intuitive thinking includes much vagueness which hinders the approach to completely rigorous proofs in analysis. In the following chapters, this will strike us more and more clearly. Even here, we may mention, for example, that the concept of a continuous curve is very difficult to grasp intuitively. A continuous curve need not by any means possess a definite direction at every point. In fact, there actually exist continuous curves, which at no point posses a direction, and continuous curves, to which no length can be assigned. In the face of such facts, even the beginner will admit the need for arithmetizing analysis.

Rigorous mathematical concepts are always very highly idealized forms of the ideas which arise intuitively. Hence it is absolutely impossible to dispose of problems relating to the ultimate foundations of mathematics by appealing to naive intuition.

Yet, we must not allow ourselves to forget that a century of brilliant and fruitful development of mathematics was possible before these requirements were fulfilled. In spite of all its defects, intuition still remains the most important driving force for mathematical discovery, and intuition alone can bridge the gap between theory and application.

We shall now follow Bolzano and Weierstrass in developing those lines of thought which yield the rigorous and complete proofs of the theorems which we have formulated by intuitive means in the first chapter.

A1.1 The Principle of the Point of Accumulation and its Applications

A1.1.1 The Principle of the Point of Accumulation: In the rigorous discussion of the fundamentals of analysis, the leading part is played by Weierstrass' principle of the point of accumulation. From an intuitive point of view, this principle is merely the statement of a triviality; however, just because it summarizes a state of affairs which occurs frequently, it is as useful as small change of money is in daily life. The principle follows:

If infinitely many numbers are given in a finite interval, these numbers possess at least one point of accumulation, that is, there is at least one point x such that in every interval about the point x, however small, there lie infinitely many of the given numbers.

In order to prove this principle arithmetically, we assume to begin with that the given interval is the interval from 0 to 1. We now subdivide this interval into ten equal parts by means of the points 0.1, 0.2, ··· , 0.9. At least one of these subintervals must contain infinitely many points. Let the interval beginning with the number 0.a₁ be that interval (or one of those intervals if there are several). We now subdivide this interval into ten parts by means of the points of subdivision 0.a₁1, 0.a₁2, ··· , 0.a₁9. Again, it is true that at least one of these subintervals must contain infinitely many points; let it be the subinterval beginning with the number 0.a₁a₂. We again subdivide into ten parts, note that one of these parts must contain infinitely many points and continue the process. We thus arrive at a sequence of digits a₁, a₂, a₃, ··· , each having one of the values 0, 1, 2, ··· , 9. We now consider the decimal

x = 0.a₁a₂a₃ ··· .

It is clear that this is a point of accumulation of our set of numbers. In fact, every interval, no matter how small, in the interior of which the point x lies, contains the subintervals of our system of subdivision from a certain degree of fineness onward, and these subintervals contain infinitely many numbers of the set.

If the interval under consideration instead of being the interval from 0 to 1 is, say, the interval from a to a + A, nothing essential in the above argument is changed. The point of accumulation is then represented simply by a number of the form

a + h ´ 0.a₁a₂a₃ ··· .

A1.1.2 Limits of Sequences:. These considerations throw new light upon the concept of the limit of an infinite sequence of numbers a₁, a₂, a₃, ··· , a_n, ··· . We first consider the exceptional case in which infinitely many numbers of the sequence are equal to one another and extend our definition by applying the name point of accumulation also to this point (or these points). If there are infinitely many different numbers in the sequence and if we assume that the numbers a_n of this sequence are bounded, i.e., that there is a number M such that |a_n| < M for all values of n, the numbers of the sequence form an infinite set of numbers in a finite interval, since they all lie between - M and M. Hence they must possess at least one point of accumulation (x). If there is only one point of accumulation, it is easy to show that the sequence converges and that its limit is the number x. In fact, let us mark off any small interval about the point x. If infinitely many points of the sequence were outside this interval, they would have a limit point other than x, contrary to the hypothesis, whence only a finite number of the numbers of the sequence are exterior to the interval and, by definition, the sequence approaches x. On the other hand, if there are several points of accumulation, the sequence approaches no limit. The existence of a limit and the uniqueness of the point of accumulation of a bounded sequence of numbers are therefore equivalent ideas.

The case of the non-existence of a limit is to be regarded as the rule rather than the exception. For example, the sequence with the terms a_2n = 1/n, a_2n-1 = 1 - 1/n, (n = 1, 2, ···.) has the two points of accumulation 0 and 1.

The aggregate of the positive rational numbers may be regarded as a sequence of numbers, in which the ordering by magnitude is, of course, completely destroyed. We arrive most easily at such an arrangement in a sequence by first writing down the rational numbers as shown and then running through this array as shown by the arrows, disregarding those numbers which have already been encountered (such as 2/4). The system of rational numbers obviously has all rational and irrational points as points of accumulation. It therefore forms a simple example of a sequence with an infinite number of points of accumulation.

By means of the concept of convergence, we can state the principle of the point of accumulation in a remarkable form which is often convenient for applications.

From every bounded infinite set of numbers, it is possible to choose an infinite sequence a₁, a₂, a₃, ··· which converges to a definite limit x. For this purpose, we have only to take a point of accumulation x of the given set of numbers, then select a number a₁ of the set the distance of which from x is less than 1/10, then a second number a₂ of the set the distance of which from x is less than 1/100, then a third number a₃, the distance of which from x is less than 1/1,000, and so on. We see at once that this sequence actually converges to the limit x.

A1.1.3 Proof of Cauchy's Convergence Test:. Let us now return to convergent sequences, i.e., to bounded sequences with only one point of accumulation. Cauchy's convergence test now reduces almost to a triviality. In fact, when m and n are sufficiently large, let |a_m - a_n| be arbitrarily small . Then all the numbers a_n lie in a finite interval and therefore have at least one point of accumulation x. If there were a second point of accumulation h, the distance of this point from x would be |x - h| = a, a positive quantity. Within an arbitrarily small distance from x, say, within a distance less than a/3 from x, there must be infinitely many numbers a_n and hence, in particular, infinitely many numbers a_n for which n > N, however large N is chosen. Similarly, within an arbitrarily small distance from the point h, say, within a distance less than a/3 from h, there are infinitely many numbers a_m of the sequence; in particular, infinitely many numbers a_m, for which m > N. For these values a_n and a_m, it is true that |a_m - a_n| > a/3, and this relation is incompatible with the hypothesis that for sufficiently large values of N the difference |a_n - a_m| is arbitrarily small provided that n and m are both greater than N. Hence there are not two distinct points of accumulation and Cauchy's test has been proved.

A1.1.4 The Existence of Limits of Bounded Monotonic Sequences: It is equally easy to see that a bounded monotonic increasing or monotonic decreasing sequence of numbers must possess a limit. In fact, let the sequence be monotonic increasing and x be a point of accumulation of the sequence; such a point of accumulation must certainly exist. Then x must be greater than any number of the sequence. For if a number a₁ of the sequence were equal to or greater than x, every number a_n for which n > l + 1 would satisfy the inequality a_n > a_l+1 > a_l ³ x. Hence all numbers of the sequence, except at most the first (l + 1), would lie outside the interval of length 2(a_l+1-x) the mid-point of which is at the point x. However, this contradicts the assumption that x is a point of accumulation. Hence no numbers of the sequence, and, a fortiori, no points of accumulation lie above x. Thus, if another point of accumulation h exists, we must have h < x. But if we repeat the above argument with h in place of x, we obtain x < h, which is a contradiction. Hence only one point of accumulation can exist and the convergence is proved. Naturally, an argument exactly analogous to this one applies to monotonic decreasing sequences.

As in 1.6.4, we can extend our statements about monotonic sequences by including the limiting case in which successive numbers of the sequence are equal, to one another. It is in this case better to speak of monotonic non-decreasing and monotonic non-increasing sequences, respectively. The theorem about the existence of a limit remains valid for such sequences.

A1.1.5 Upper and Lower Points of Accumulation; Upper and Lower Bounds of a Set of Numbers: In the construction in A1.1.1, which led us to a point of accumulation x, we must at each step coose a subinterval containing infinitely many points of the set. Had we always chosen the last subinterval which contained an infinite number of points, we should have been led to a certain definite point of accumulation b. This point of accumulation b is called the upper point of accumulation or the upper limit of the set of numbers and is denoted by It is that point of accumulation of the sequence which lies furthest to the right, i.e., it is quite possible that an infinite number of points of the sequence lie above b, but no matter how small the positive number e may be, there are not an infinite number above b + e.

If we had always chosen in the construction of A1.1.1 the first of the intervals containing an infinite number of points of the set, we should again have arrived at a certain definite point of accumulation a. This point a is called the lower point of accumulation or lower limit of the set and is denoted by There may be infinitely many numbers of the set below a, but, no matter how small is the positive number e, there are only a finite number below a - e. The proofs of these facts can be left to the reader.

Neither the upper limit b nor the lower limit a need belong to the set. For example, for the set of numbers a_2n = l/n, a_2n-1 = 2 - l/n, these limits are a = 0 and b = 2, respectively, but the numbers 0 and 2 do not themselves occur in the set.

There is in this example no number of the set above b = 2. In this case, we say that b = 2 is also the upper bound M of the set, according to the following definition: M is called the least upper bound or simply the upper bound of a set of numbers, if: (1) there is no number of the set greater than M, but (2) there is a number of the set greater than M - e for every positive number e. The upper bound may coincide with the upper bound, as in the example above. But the set a_n = 1 + l/n (n = 1, 2 ···) shows that this is not necessarily the case. Here M = 2 and b = 1.

Every bounded set of numbers has a least upper bound. In fact, let b be the upper limit of the set. Either there are no numbers of the set larger than b or there are such numbers. In the first case, b is the least upper bound, since no numbers are above b and there are numbers arbitrarily close to b below it. In the second case, let a be a number of the set greater than b. There are only a finite number of numbers of the set equal to or greater than a, since otherwise there would be a point of accumulation above b, which is impossible. We therefore need only choose the greatest of these numbers; it will be the upper bound of the set.

We see that in any case M ³ b and we recognize the fact: If the upper bound of a set does not coincide with the upper limit, it must belong to the set and is an isolated point of the set.

Corresponding statements hold for the lower bound m; it is always equal to or less than a, and if m and a do not coincide, m belongs to the set and is an isolated point of the set.

A1.2. Theorems on Continuous Functions

A1.2.1. Greatest and Least Values of Continuous functions: A bounded, infinite set of numbers must possess a least upper bound M and a largest lower bound m. However, as we have seen, these numbers M and m do not necessarily belong to the set; as we say: The set does not necessarily have a greatest or a least value.

In view of this fact, the following theorem on continuous functions is by no means as obvious as it appears to be to simple intuition: Every function f(x), which is continuous in a closed interval a £ x £ b, assumes at least once a greatest and a least value or, as we say, it possesses a greatest and a least value.

This may easily be proved as follows: The values assumed by the continuous function f(x) in the interval a £ x £ b form a bounded set of numbers and therefore possess a least upper bound M. Otherwise, a sequence of numbers x₁, , x₂, ··· , x_n, ··· in our interval would exist for which f(x) increases beyond all bounds. This sequence would have at least one point of accumulation in the interval. In that case, arbitrarily near to , there would always be numbers x_n of our sequence for which the expression |f() - f(x_n| exceeds 1 (and, in fact, is arbitrarily large), that is, the function would be discontinuous at the point . Thus, a least upper bound M exists and hence either there is a point x such that f(x ) = M, which would prove the statement, or there is a sequence of numbers x₁, x₂, ··· , x_n, ··· in the interval for which

According to the formulated principle of the point of accumulation, we can select a subsequence of the numbers x_n„ which converges to a limit x. Let us call this subsequence x₁, x₂, ··· , x_n, ···, so that

It is then certain that

Ott the other hand, the function has been assumed to be continuous in the interval and, in particular, at x, whence

Hence f(x) = M. The value M is therefore assumed by the function at a definite point x in the interior or on the boundary of the interval, as stated. An exactly similar discussion applies to the least value.

In general, the theorem about the greatest and least values of continuous functions does not remain true unless we expressly assume the interval to be closed, that is, unless we make the hypothesis of continuity also refer to the end-points. For example, the function y = 1/x is continuous in the open interval 0 < x < ¥. However, it has no greatest value, but has arbitrarily large values near x = 0. Similarly, it has no least value, but comes arbitrarily near 0 for sufficiently large values of x without ever assuming the value 0.

A1.2.2 The Uniformity of Continuity: As we have already seen and as we shall further see, the continuity of a function f(x) in a closed interval a £ x £ b leaves room for a variety of possibilities, which do not suggest themselves intuitively. For this reason, we shall give logically rigorous proofs of certain consequences of the idea of continuity which, from a naïve point of view, seem to be quite obvious. The definition of continuity simply states that there follows from the relation

the relation

We can also express this as follows: There correspond to every e > 0 for each point x a number d > 0 such that |f(x) - f(x)| < e whenever |x - x| < d, provided that all the numbers x considered lie in the interval a £ x £ b.

For example, in the case of the function y = cx (where c ¹ 0), such a number d is given by the relation d = e |c|. For the function y = x², we can find such a number as follows: We assume that a=0 and b=1, and ask ourselves how near to x the number x must lie in order that the expression |x² - x ²| may be less than e. For this purpose, we write

Hence, if we choose d £ e /(1 + x), we can be sure that |x² - x ²| < e. We see in this example that the number d found in this way depends not only on e, but also on the point of the interval at which we are investigating the continuity of the function. However, if we give up the attempt to make the best possible choice of d for each x, we can eliminate this dependence of d on x. In fact, we need only replace x on the right hand side by the number 1 and thus obtain for d the expression e /2, which is smaller than the previous expression for d, but serves equally well for all points x.

There arises now the question whether something similar does not hold for every function which is continuous in a closed interval, that is, we enquire whether it may not be possible to determine for each e a d = d(e) which depends only on e and not on x such that the inequality

holds, provided |x - x| < d for all values of x at the same time (or, better expressed, uniformly with respect to x). As a matter of fact, this is possible merely as a consequence of the general definition of continuity without any additional assumptions. This fact, which first attracted attention late in the Nineteenth Century, is called the Theorem of the Uniform Continuity of Continuous Functions.

We shall prove this theorem indirectly, that is, we shall show that the assumption that a function f(x) exists, which in a closed interval a £ x £ b is continuous and yet not uniformly continuous, leads us to a contradiction. Uniform continuity means that, if we wish to make the difference |f(u) - f(v)| less than an arbitrarily chosen positive number e, the numbers u and v being chosen in the closed interval a £ x £ b, we need only choose u and v near enough to one another, namely, at a distance apart which is less than d = d(e); it is immaterial where in the interval the pair of numbers u,v is chosen. Now, if f(x) were not uniformly continuous, there would exist a positive (perhaps very small) number a with the property: There corresponds to every number d_n of an arbitrary sequence d₁, d₂, ··· of positive numbers tending to zero a pair of values u_n,v_n of the interval for which |u_n - v_n| < d_n and |f(u_n) - f(v_n)| > a. According to the principle of the point of accumulation, the numbers u_n must have a point of accumulation x and the numbers v_n must have the same point of accumulation. If we select an arbitrarily small interval |x - x| < d about this point x , an infinite number of the pairs u_n,v_n will lie in this interval. But this contradicts the assumed continuity of f(x) at the point x ; in fact, by Cauchy's convergence test, this requires that the points x₂ and x₁ are near enough to x

The uniformity of the continuity is thus proved.

In our proof, we have made essential use of the fact that the interval is closed*. In fact, the theorem of uniformity of continuity does not hold for intervals which are not closed.

* 0therwise the point of accumulation need not belong to the interval.

For example, the function l/x is continuous in the half-open interval 0 < x £ 1, but it is not uniformly continuous, because no matter how small the length d (< l) of an interval is chosen, the function will take values differing by any fixed number, say 1, in the interval, if only the interval lies near enough to the origin, say, d /2 £ x £ 3d /2. Of course, the non-uniformity of continuity is due to the fact that in the closed interval 0 £ x £ 1the function possesses at the origin a discontinuity. If we had considered the example y = x² in the entire (open) interval - ¥ < x < ¥ instead of in a closed interval, it would not have been continuous.

A1.2.3 The Intermediate Value Theorem: There is another theorem, which constantly recurs in analysis:

A function f(x), continuous in a closed interval a £ x £ b, which is negative for x = a and positive for x = b (or conversely), assumes the value 0 at least once in the interval.

Geometrically speaking, this theorem is trivial, since it merely states that a curve, which begins below the x-axis and ends above it, must cut the axis somewhere in between. Analytically speaking, the theorem is very easily proved. There are in the interval an infinite number of points for which f(x) < 0; in fact, due to the continuity of the function, this is true for an entire interval beginning at the point a. The set consisting of those points x for which f(x) < 0 has a least upper bound x which is greater than a. Since there are points in every neighbourhood of x for which f(x) < 0, we must have f(x) £ 0 (whence, in particular, x ¹ b). However, it is impossible that f(x) £ 0, (whence, in particular, x ¹ b). However, it is impossible that f(x) < 0, because then f(x) would be negative in a sufficiently small neighbourhood of x, including values of x > x , in contradiction to the assumption that x is the upper bound of the values of x for which f(x) < 0. Hence f(x) = 0 and our assertion is proved.

A slight generalization of our theorem is: If we assume that f(a)=a and f(b)=b and if m is any value between a and b, the continuous function f(x) assumes the value f()m at least once in the interval, because the continuous function

will have different signs at the two ends of the interval and will therefore assume the value 0 somewhere inside it.

A1.2.4 The Inverse of a Continuous Monotonic Function: If the continuous function y = f(x) is monotonic in the interval a £ x £ b, it will assume each value between f(a) and f(b) once and only once; hence, if y describes the closed interval between the values a = f(a) and b = f(b), there will correspond to each value of y exactly one value of x. We can therefore think of x as a single-valued function of y in this interval, i.e., the function y = f(x) has a unique inverse. We assert that this inverse function x = f(y) is also a continuous, monotonic function of y, as y varies within the interval between a and b.

The monotonic character of the inverse function x =f (y) is obvious. In order to prove its continuity, we observe that it follows from the monotonic character of f(x) that

provided that x₁ and x₂ are distinct numbers of the interval. If h is a positive number less than b - a, the function

is continuous in the closed interval a £ x £ b - h,whence it has a least value

which by our preceding remark is not zero.* We conclude from this that, if x₁ and x₂ are two points in the interval for which |x₁ - x₂| ³ h, then

However, this implies the continuity of the inverse function, because, if |y₁ - y₂| falls below the positive number a (h), then we must have |x₁ - x₂| ³ h < h, whence, if a positive number e is given, we need only choose d equal to a(e) in order to ensure that for all values y for which |y₁ - y₂| < d it is also true that |f(y₁) - f(y₂)| < e.

* On account of the continuity of f(x), naturally, a (h) itself tends to 0 with h.

Hence we have established the theorem: If the function y = f(x) is continuous and monotonic in the interval a £ x £ b and f(a) = a, f(b) = b, then it has a single-valued inverse function x = f (y), a £ y £ b and this inverse function is also continuous and monotonic.

A1.2.5 Further Theorems on Continuous Functions: We leave it to the reader to prove the following almost trivial fact: A continuous function of a continuous function is itself continuous, i.e., if f (x) is a function, continuous in the interval a £ x £ b, its functional values lie in the interval a £ f £ b and, in addition, if f(f) is a continuous function of f in this last interval, then f(f (x)) is a continuous function of x in the interval a £ x £ b. (Theorem of the continuity of functions of continuous functions.)

It has already been mentioned that the sum, difference and product of continuous functions are themselves continuous and that the quotient of continuous functions is continuous, provided that the denominator remains different from zero.

A1.3 Some Remarks on the Elementary Functions:

In Chapter I, we have tacitly assumed that the elementary functions are continuous. The proof of this fact is very simple. Firstly, the function f(x) = x is continuous, whence y = x² as the product of two continuous functions is continuous and every power of x is likewise continuous. Thus, every polynomial is continuous, being a sum of continuous functions. Every rational, fractional function, as a quotient of continuous functions, is likewise continuous in every interval in which the denominator does not vanish.

The function xⁿ is continuous and monotonic. Hence the n-th root, as the inverse function of the n-th power, is continuous. By the theorem of the continuity of functions of continuous functions, the n-th root of a rational function is continuous (except where the denominator vanishes).

The continuity of the trigonometric functions, with which the reader is familiar from elementary mathematics, could now readily be proved, using the concepts developed above. The discussion is not given here, since it will be seenin 2.3.5 that this continuity follows naturally as a consequence of their differentiability.

We shall merely comment here on the definition and continuity of the exponential function a^x, the general power function x^a and the logarithm. We assume, as in 1.3.4, that a is a positive number, say, greater than 1, and, if r=p/q is a positive rational number (p and q being integers), we take a^r = a^p/q to mean the positive number the q-th power of which is a^p. If a is any irrational number and r₁, r₂, ··· , r_m, ··· is a sequence of rational numbers approaching a, we assert that exists and then call this limit a^a.

In order to prove the existence of this limit by Cauchy's test, we need only show that is arbitrarily small, provided that n and m are sufficiently large. For example, let r_n > r_m, i.e., that r_n - r_m = d, where d > 0. Then

Since remains bounded, we need only show that

is arbitrarily small when the values of n and m are sufficiently large. However, d is a rational number and certainly may be made as small as we please, provided the values of n and m are sufficiently large. Hence, if l is an arbitrarily large positive integer, d < l/l, if n and m are large enough. Now, the relations d < l/l and a > 1 yield*

and since a^1/l tends to l as l increases our assertion follows immediately.

This statement follows from the fact that, when a > 1, the power a^m/n is greater than 1 provided m/n is positive. This is clearly true. For if a^m/n were less than 1, then a^m = (a^m/n)ⁿ would be a product of n factors all less than 1 and would be less than 1. On the contrary, a^m is the product of n factors all greater than 1, and so it is greater than 1.

It may be left to the reader to show that the function a^x, extended to irrational values in this manner, is also continuous everywhere and, moreover, that it is a monotonic function. For negative values of x, this function is naturally defined by the equation

As x runs from - ¥ to +¥, a^x takes all values between 0 and +¥. Consequently, it possesses a continuous and monotonic inverse function, which we call the logarithm to the base a. In a similar manner, we can prove that the general power x^a is a continuous function of x, where a is any fixed rational or irrational number and x varies over the interval 0 < x < ¥, as well as that x^a is monotonic if a ¹ 0.

The elementary discussion of the exponential function, the logarithm and the power x^a, which has been sketched here will later on be replaced by another discussion which, in principle, is much simpler.

Exercises 1.8:

1. Give the upper and lower bounds and upper and lower limits for the following sequences and state which belong to the sequence:

2.* Prove that if f(x) is continuous for a £ x £ b, then there exists for every e > 0 a polygonal function f (x) (i.e., a continuous function the graph of which consists of a finite number of rectilinear segments meeting at corners} such that |f(x) - f (x)| < e for every x in the interval.

3. Prove that every polygonal function f (x) can be represented by a sum

where the x_i are the abscissae of the corners.
Find a formula of this kind for the function f(x), defined by the equations:

4. As in A1.2.2, find for the following functions d (e) such that (f(x₁) - f(x₂)| < d(e):

5.* The function y = sin (1/x) has a discontinuity in the interval .0 < x < 1. Prove that it is not uniformly continuous in that open interval.

6. A function f(x) is defined for all values of x as follows:

f(x) = 0 for all irrational values of x,
f(x) = 1/q for all rational x = p/q,

where p/q is a fraction in its lowest terms (thus, for x = 16/29, f(x) = 1/29).

Prove that f(x) is continuous for all irrational and discontinuous for all rational values of x.

Answer and Hints

Appendix II to Chapter I

A2.1 Polar Co-ordinates: In Chapter I, we have placed the concept of function in the foreground and represented functions geometrically by means of curves. However, it is useful to recall that analytical geometry follows the reverse procedure; it begins with a curve given by some geometrical property and represents this curve by a function, for example, by a function which expresses one of the coordinates of a point of the curve in terms of the other coordinate. This point of view naturally leads us to consider, apart from rectangular coordinates, to which we have restricted ourselves in Chapter I, other systems of coordinates which may be better suited for the representation of curves which are given geometrically, The most important example is that of the polar coordinates r, q, connected with the rectangular co-ordinates x, y of a point P by the equations

and the geometrical interpretation of which follows from Fig. 25.

Consider as an example the lemniscate. It is geometrically defined as the locus of all points for which the product of the distances r₁ and r₂ from the fixed points P₁ and P₂ with the rectangular co-ordinates x = a, y = 0 and x = -a, y=0, respectively, has the constant value a² (Fig. 26). Since

a simple calculation yields the equation of the lemniscate in the form

If we now introduce polar coordinates, we obtain

now, division by r² and use of a simple trigonometric formula yields

Thus, we see that the equation of the lemniscate in polar coordinates is simpler than in rectangular coordinates.

A2.2. Remarks on Complex Numbers: Our study will be based chiefly on the class of real numbers. Nevertheless, with a view to the discussions in Chapters VIII, IX and XI, we remind the reader that the problems of algebra have led to a still wider extension of the number concept, namely, to the introduction of complex numbers. The advance from the natural numbers to the class of all real numbers arose from the desire to eliminate exceptional phenomena and to make always possible operations such as subtraction, division, and correspondence between points and numbers. Similarly, we are compelled, by the requirement that every quadratic equation and, in fact, every algebraic equation shall have a solution, to introduce complex numbers. For example, if we wish the equation

to have roots, we are obliged to introduce the new symbols i and -i as the roots of this equation. (It is shown in algebra that this is sufficient to ensure that every algebraic equation shall have a solution.*

* That every algebraic equation possesses real or complex roots is the statement of the Fundamental Theorem of Algebra.

If a and b are ordinary real numbers, the complex number c = a + ib denotes a pair of numbers (a, b), calculations with such pairs of numbers being performed according to the general rule: We add, multiply and divide complex numbers (which include the real numbers as the special case b = 0), treating the symbol i as an undetermined quantity and then simplify all expressions by using the equation i² = -1 to remove all powers of i higher than the first, thus leaving only an expression of the form a + ib.

We may assume that the reader has already a certain degree of familiarity with these complex numbers. We shall nevertheless emphasize a particularly important relationship which we shall explain in connection with the geometrical or trigonometrical representation of complex numbers. If c = x + iy is such a number, we represent it in a rectangular co-ordinate system by the point P with the co-ordinates x and y. By means of the above equations x= r cos q, y=r sinq, we now introduce the polar coordinates r and q instead of the rectangular co-ordinates x and y. Then is the distance of the point P from the origin and q is the angle between the positive x-axis and the segment OP. The complex number c is now represented in the form

The angle q is called the amplitude of the complex number c, the quantity r its absolute value or modulus) for which we also write |c|Id. Obviously, there corresponds to the conjugate complex number c = x - iy the same absolute value, but (except in the case of negative real values of c) the angle - q. Obviously,

If we use this trigonometrical representation, the multiplication of complex numbers takes a particularly simple form, because then

If we recall the addition theorems for the trigonometric functions, this becomes

Hence we multiply complex numbers by multiplying their absolute values and adding their amplitudes. The remarkable formula

is usually called De Moivre's Theorem. It leads us at once to the relation

which, for example, enables us at once to solve the equation xⁿ = 1 for positive integers n, the roots (the so-called roots of unity) being

Moreover, if we imagine the expression on the left-hand side of the equation

expanded by the binomial theorem, we need only separate the real and imaginary parts, in order to obtain expressions for cos nq and sin nq in terms of powers and products of powers of sin q and cos q.

Exercises 1.9:

1. Plot the graphs of the functions:

2. Find the polar equation of

(a) the circle with radius a with centre at the origin,
(b) the circle with radius a with centre (a, f₀);
(c) the general straight line.

3. Use De Moivre's theorem to express cos 2q and sin 2q in terms of sin q and cos q ; similarly, for cos 3q, sin 3q, cos 5q, sin 5q.
Prove that cos nq is a polynomial in cos q, and also that, if n is odd, sin nq is a polynomial in sin q.
4. Work out the following expressions and state the modulus and amplitude of each of the numbers involved and of the answers:

6.* Prove that, if where n is an integer greater than 1,

Answers and Hints

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