A2.1. THE PRINCIPLE OF THE POINT OF ACCUMULATION IN SEVERAL DIMENSIONS AND ITS APPLICATIONS
If we wish to refine the concepts of the theory of functions of several variables and to establish it on a firm basis, without reference to intuition, we proceed in exactly the same way as in the case of functions of one variable. It is sufficient to discuss these matters in the case of only two variables, since the methods are essentially the same for functions of more than two independent variables.
A2.1.1 The Principle of the Point of Accumulation: We again base our discussion on Bolzano and Weierstrass'principle of the point of accumulation. A pair of numbers (x, y) will be called a point P in a space of two dimensions and may be represented in the usual way by means of a point with the rectangular co-ordinates x and y in an xy-plane. We now consider a bounded infinite set of such points P(x, y), that is, the set is to contain an infinite number of points and all the points are to lie in a bounded part of the plane, so that |x| < C and | y| < C, where C is a constant. The principle of the point of accumulation can then be stated as follows: Every bounded infinite set of points has at least one point of accumulation. Thus, there exists a point Q with co-ordinates (x, h) such that an infinite number of points of the given set lie in every neighbourhood of the point Q, say, in every region
where d is any positive number. Or, in other words, out of the infinite set of points we can choose a sequence P1, P2, P3, ··· . in suck a way that these points approach a limit point Q.
This principle of the point of accumulation is just as intuitively clear for several dimensions as it is for one dimension. It can be proved analytically by the method, used in the corresponding proof in Volume 1, merely by substituting rectangular regions for the intervals used there. However, an easier proof can be constructed by using the principle of the point of accumulation for one dimension. In order to do this, we note that, by assumption, every point P(x, y) of the set has an abscissa x for which the inequality |x| < C holds. Either there is an x = x0 which is the abscissa of an infinite number of points P (which therefore lie vertically above each other) or else each x belongs only to a finite number of points P.
In the first case, we fix upon x0 and consider the infinite number of values of y such that (x0, y) belongs to our set. These values of y have a point of accumulation h0, by the principle of the point of accumulation for one dimension. Hence, we can find a sequence of values of y, say y1, y2, ··· such that yn ® h0, from which it follows that the points (x0, yn) of the set tend to the limit point (x0, y0), which is thus a point of accumulation of the set.
In the second case, there must be
an infinite number of distinct values of x which are the
abscissae of points of the set and we can choose a sequence x1,
x2, ··· of these abscissae tending to
a unique limit x . For each xn,
let Pn,(xn,
yn) be a point of the set with abscissa xn.
The numbers yn are an infinite bounded
set of numbers, so that we can choose a sub-sequence 
. tending to a limit h.
The corresponding sub-sequence of abscissae 
still
tends to the limit x , whence the points 
tend
to the limit point (x, h). Hence. in either case, we can
find a sequence of points of the set tending to a limit point and
the theorem is proved.
A first and important consequence
of the principle of the point of accumulation is Cauchy's convergence test, which can be expressed as
follows: A
sequence of points P1, P2, P3···with
the co-ordinates (x1, y1), (x2,
y2), ··· tends to a limit point if, and only
if, there is for every e > 0 a subscript N = N(e)
such that the distance 
between the points Pn and Pm
is less than ee whenever both n and m are
larger fhan N.
A2.1.2. Some Concepts of the Theory of Sets of Points:. The general concept of a limit point is fundamental in many of the more refined investigations of the foundations of analysis, based on the theory of sets of points. Although these matters are not essential for most of the purposes of this book, we shall mention some of them here for the sake of completeness.
A bounded set of points, consisting of an infinite number of points, is said to be closed if it contains all its limit points, i.e., limit points of sequences of points of the set are again points of the set. For example, all the points lying on a closed curve or surface form a closed set. For functions, defined in closed sets, we can state the two fundamental theorems:
A function which is continuous in a bounded closed set of points assumes a greatest (and a smallest value in that set.
A function which is continuous in a bounded closed set is uniformly continuous in that set.
The proofs of these theorems are so like the corresponding proofs for functions of one variable that we shall omit them.
The least upper bound of the distance between the points P1 and P2 for all pairs of points P1, P2, where both points belong to a set, is called the diameter of that set. If the set is closed, this upper bound will actually be assumed for a pair of points of the set. The student will be able to prove this easily, remembering that the distance between two points is a continuous function of the co-ordinates of the points.
By using the theorem that a continuous function on a bounded closed set does assume its least value, we can readily establish the following fact: If a point P does not belong to a closed set M, there exists a positive least distance from P to M, i.e., there exists a point Q of M such that no point of M has a smaller distance from P than Q. This enables us to show that the closed regions defined in 2.1.1 are actually closed sets according to the present definition. In fact, let C be a closed curve and R the closed region consisting of all internal points of C or on C'; we must show that all the limit points of R belong to R. We assume the opposite, i.e., that there is a point P not belonging to R which is a limit point of R. Then, in particular, P does not lie on C, whence, by the theorem above, it has a positive least distance from C (C being a closed set). We can therefore describe a circle about P as centre so small that no point of C lies in the circle; we need only make the radius of the circle less than the smallest distance between P and C. The point P is outside C, since otherwise it would belong to R; and since every point in the small circle can be joined to P by a line-segment which does not cross the curve C, every point of the circle lies outside C, whence no point of the circle belongs to B. But we have assumed that P is a limit point of R, which requires that the circle should contain an infinite number of points of R. Hence the assumption that there is a limit point of R which does not itself belong to R leads to a contradiction and the assertion is proved. The extension to closed regions R bounded by several closed curves is obvious.
A useful property of closed sets is contained in the theorem on shrinking sequences of closed sets :
If all the sets M1, M2, M3, . . . are closed, and each set is contained in the preceding one, then there exists a point (x, h) which belongs/ to all the sets.
Choose in each of the sets Mn
a point Pn. The sequence Pn
must either contain an infinite number of repetitions of someone
point or else an infinite number of distinct points. If one point
P is repeated an infinite number of times, then. it
belongs to all the seta; in fact, if Mn
is any one of the sets, P belongs to a set 
,
where n1 > n and 
is
contained in Mn. If there are an
infinite number of distinct points Pn,
then they possess by the principle of the point of accumulation a point of accumulation (x , h).
This point belongs to each Mn,
because. whenever m > n, the point Pm
belongs to Mn, since it is a
point of Mm which is contained in Mn.
Hence (x , h) is a limit point of points Pm
of Mn, and since Mn
is closed, (x , h) is a point of Mn.
Thus in either case there exists a point common to all the sets Mn
and the theorem is proved.
The assumption that the sets Mn are closed is essential, as the following example shows. Let Mn be the set 0 < x < 1/n. Each set is contained in the preceding one, but no point belongs to all the sets. In fact, if x = 0, the point belongs to no set, while if x > 0 it belongs to no set Mn for which 1/n < x.
A set is said to be open if we can find for every point of the set a circle about the point as centre which belongs completely to the set. An open set is connected if every pair of points A and B of the set can be joined by a broken (polygonal) line which lies entirely in the set. The word domain
The word domain is often used with the restricted meaning of a connected open set. As examples, we have the boundary points.The boundary B of a domain D is a closed set. We shall sketch the proof of this statement. A point P which is a limit point of B does not belong to D, because every point of D lies in a circle composed only of points of D and hence is devoid of points of B. It is also a limit point of D, because we can find arbitrarily close to P a point Q of B, and arbitrarily close to Q points of D. Hence P belongs to B.
If we add to a domain D its boundary points B, we obtain a closed set, because every limit point of the combined set is either a limit point of B and belongs to B or is a limit point of D and belongs either to D or to B. Such sets are said to be closed regions, and are particularly useful for our purposes.
Finally, we define a neighbourhood of a point P as any open set containing P. If we denote the co-ordinates of P by (x, h), the two simplest examples of neighbourhoods of P are the circular neighbourhood, consisting of all points (x, y) such that
and the square neighbourhood, consisting of all points (x, y) such that
A2.1.3 The Heine-Borel Covering Theorem: A further consequence of the principle of the point of accumulation, which is useful in many proofs and refined investigations, is the Heine-Borel covering theorem:
If corresponding to every point of a bounded closed set M a neighbourhood of the point, say, a square or a circle, is assigned, it is possible to choose a finite number of these neighbourhoods in such a way that they completely cover M.
Naturally, the last statement means that every point of M belongs to at least one of the finite number of selected neighbourhoods. By an indirect method, the proof can be derived almost immediately from the theorem on shrinking closed sets. Suppose that the theorem is false. The set M, being bounded, lies in a square Q. Subdivide this square into four equal squares. Because at least one of these four squares, the part of M lying in or on the boundary of that square, cannot be covered by a finite number of the neighbourhoods; in fact, if each of the four parts of M could be covered in this way, M itself would be covered. This part of M we call M1 and we see at once that M1, is closed.
Subdivide now the square containing M1 into four equal squares. By the same argument, the part M2 of M1 lying in or on the boundary of one of these squares cannot be covered by a finite number of the neighbourhoods. Continuing this process, we obtain a sequence of closed sets M1, M2, M3, ··· each enclosed in the preceding one; each of these is contained in a square the side of which tends to zero, and none of them can be covered by a finite number of the neighbourhoods. By the theorem on shrinking sequences of closed sets, we know that there is a point (x, h) which belongs to all these sets, and hence, a fortiori, it belongs to M. Thus, there corresponds to the point (x, h) one of the neighbourhoods containing a small square about (x , h). But since each Mn contains (x ,h) and is itself contained in a square the side of which tends to 0 as does 1/n, each Mn after a certain n is completely contained in the small square about (x, h) and is therefore covered by one neighbourhood of the set. The assumption that the theorem is false has therefore led to a contradiction and the theorem is proved.
1. A convex region B may be defined as a bounded and closed region with the property that if A, B are any two points belonging to R, all points of the segment AB belong to R. Prove the statements:
(a)* If A is a point not belonging to R, there exists a straight line passing through A which has no point in common with R.
(b)* There exists through every point P on the boundary of R a straight line l (a so-called line of support) such that all points of R lie on one and the same side of l or on l itself.
(c) If a point A lies on the same side of every line of support as the points of R, then A is also a point of B.
(d) The centre of mass of R is a point of R.
(e) A closed curve forms the boundary of a convex region, provided that it has not more than two points in common with any straight line.
(/)* A closed curve forms the boundary of a convex region, provided that its curvature is everywhere positive. (It is assumed that if the entire curve is traversed the tangent makes one complete revolution.
2. (a) If S is an arbitrary closed and bounded set, there is one least convex envelope E of S, i.e., a set which (1) contains all points of S, (2) is contained m all convex sets containing S, (3) is convex,
(b) E may also be described in the following way: A point P is in E if, and only if, for every straight line which leaves all points of S on one and the same side, P is also on this side.
(c) The centre of mass of S is a point of E.
A2.2 The Concept of Limit for Functions of Several Variables
We shall find it useful to refine our concepts of the various limiting processes connected with several variables and to consider them from a single point of view. Here we again restrict ourselves to the typical case of two variables.
A2.2.1 Double Sequences and their Limits:. In the case of one variable, we began with the study of sequences of numbers an, where the suffix n could be any integer. Here double sequences have a corresponding importance. These are sets of numbers anm with two subscripts, which run through the sequence of all integers independently of each other, so that we have, for example, the numbers
Examples of such sequences are the sets of numbers
We now make the statement:
The double sequence anm converges as n and m to a limit, or more precisely to a double limit l, if the absolute differences |anm - l| is less than an arbitrarily small pre-assigned positive number e whenever n and m are both sufficiently large, i.e., whenever they are both larger than a certain number N depending only on e.
We then write
Thus, for example,
Following Cauchy, we can determine, without referring to the limit, whether the sequence converges or not by using the criterion:
The sequence anm converges if, and only if, there exists for every e > 0 a number N =N(e) such that |anm - an' m'| < e whenever the four subscripts n, m, n', m' are all larger than N.
Many problems in
analysis which involve several variables depend on the resolution
of these double
limiting processes into two successive ordinary limiting
processes. In other words, instead of allowing n and m
to increase simultaneously beyond all bounds, we first attempt to
keep one of the suffixes, say m, fixed, and let n
alone tend to . The limit thus found (if it exists) will, in
general, depend on m, let us say that it has the value l
. We now let m tend to o . There now arises the
question whether, and if so when the limit of l is
identical with the original double limit, and also the question
whether we obtain the same result no matter which variable we
first allow to increase, that is, whether we could have first
formed the limit 
and then the limit 
and
still have obtained the same result.
We shall
begin by gaining a general idea of the situation from a few
examples. In the case of the sequence anm=1/(n+m), when m
is fixed, we obviously obtain the result 
whence
the same result is obtained if we execute the limits
in reverse order. However, for the sequence
we obtain
and consequently
on performing the passages to the limit in the reverse order, we first obtain
and then
In this case, the result of the successive limiting processes is not independent of their order:
In addition, if we let n
and m increase beyond all bounds simultaneously, we find
that the double limit fails to exist. In fact, if such a limit
existed, it would necessarily have the value 0, since we can make
anm arbitrarily close to 0 by choosing n
large enough and m = n˛. On the other hand, anm
= 1/2 whenever n = m, no matter how
large is n. These two facts contradict the assumption that
the double limit exists. But even when 
, the double
limit 
may fail to exist, as is shown by the example 
Another example is the sequence
Here, the double
limit exists and has the value 0, since the absolute value
of the numerator of the fraction can never exceed 1, while the
denominator increases beyond all bounds. We obtain the same
limit, if we first let m tend to m1; we
find that 
so that 
. However, if we wish to
perform the passages to the limit in the reverse order, keeping m
fixed and letting n increase beyond all bounds, we
encounter the difficulty that lim sin n does not exist.
Hence the resolution of the double limit process into two
ordinary limiting processes cannot be carried out in both ways.
This situation can be summarized by means of two theorems. The first of these is:
If the double limit 
exists and the simple limit 
exists for every value
of m, then the limit 
also exists, and
Again,
if the double limit exists and has the value l, and the
limit
exists for every value of n, then 
also
exists and has the value l. In symbols, we have
the double limit can be resolved into simple limiting processes and this resolution is independent of the order of these limiting processes.
The proof
follows almost at once from the definition of the double limit.
By virtue of the existence of 
there exists for every
positive e an N= N(e) such that the
relations |anm - l|<e holds whenever n and m
are both larger than N. If we now keep m fixed and
let n increase beyond all bounds, we find that
This inequality holds
for any positive e provided only that m is larger
than N(e); in other words, it is equivalent to the
statement 
. The other part of the theorem
can be proved .in a similar way.
The second theorem is in some respects a converse of the first one. It gives a sufficient condition for the equivalence of a repeated limiting process and a double limit. This theorem is based on the concept of uniform convergence, which we define as follows:
The
sequence anm converges to the limit lm
uniformly in m, provided that the limit 
exists for every m and, in addition, for every positive e,
it is possible to find. an N = N(e),
depending on e, but not on m, such that |lm-anm|
< e whenever n >N.
For example, the sequence
converges uniformly to the limit lm = 1/m, as we see immediately from the estimate
we need only set N
. On the other hand, the condition for uniform convergence does
not hold in the case of the sequence anm
= m/(m + n). For fixed values of m, the
equation 
is always true; but the convergence is not uniform,
because if any particular value, say 1/100, is assigned to e,
then, no matter how large a value of n we choose, there
are always values of m for which |anm
-lm|= anm
exceeds e. We need only take m = 2n in order
to obtain anm = 2/3, which
is a value differing from the limit 0 by more than 1/100.
We now have the theorem:
If the limit anm exists uniformly
with respect to m and, moreover,
the limit 
exists, then the double limit 
exists
and has the value l:
We can
then reverse the order of the passages to the limit, if 
exists.
Use of the inequality
allows to carry out the proof as for the previous theorem, whence we leave it to the reader.
A2.2.2 Double Limits in the Case of Continuous Variables: In many cases, there occur limiting processes in which certain subscripts, for example, n are integers and increase beyond all bounds, while at the same time one or more continuous variables x, y,··· tend to limiting values x, h, ···. Other processes involve only continuous variables and no subscripts. Our previous discussions apply to such cases without essential modifications. To start with, we point out that the concept of the limit of a sequence of functions fn(x) or fn(x, y) as n can be classified as one of these limiting processes. We have already seen in 8.4.3 of Volume 1 that the definition and proofs can be applied without change to functions of several variables, so that, if the convergence of the sequence fn(x) is uniform, the limit function f(x) is continuous, provided that the functions fn(x) are continuous. This continuity yields the equations
which express the reversibility of the order of the passages to the limit n and x .
Further examples of the part played by the question of the reversibility of the order of the passages to the limit have already occurred, e.g., in the theorem on the order of partial differentiation, and we encounter other examples below.
We mention here only the case of the function
For fixed
non-zero values of y, we obtain the limit 
while we have for fixed non-zero
values of x
.
Thus,
and the order of the passages to the limit is not immaterial. This is, of course, linked to the discontinuity of the function at the origin.
In conclusion, we note that for continuous variables the resolution of a double limit into successive ordinary limiting processes and the reversibility of the order of the passages to the limit are controlled by theorems which correspond exactly to those established above for double sequences.
A2.2.3 Dini's Theorem on the Uniform Convergence of Monotonic Sequences of Functions: In many refined analytical investigations, it is useful to be able to apply a certain general theorem on uniform convergence, which we shall now state and prove. We already know that a sequence of functions may converge to a continuous limit function even though the convergence is not uniform. However, in an important special case, we can conclude from the continuity of the limit that the convergence is uniform. This is the case in which the sequence of functions is monotonic, i.e., when for all fixed values of x the value of the function fn(x) either increases steadily or decreases steadily with n. Without loss of generality, we may assume that the values increase or do not decrease, monotonically; we can then state the theorem:
If in the closed region R the sequence of continuous functions fn(x, y) converges to the continuous limit function f(x, y) and if at each point (x, y) of the region the inequality
holds, then the convergence is uniform in R.
The proof
is indirect and is a typical example of the use of the principle of
the point of accumulation. If the convergence is not uniform, there
will exist a positive number a such that for arbitrarily
large values of n - say, for all the values of n
belonging to the infinite set n1, n2,
··· - the value of the function at a point Pn
in the region, fn(Pn)
differs from f(Pn) by more than a.
If we let n run through the sequence of values n1,
n2, ···, the points 
will have at least one point of
accumulation Q,
and since R is closed, Q will belong to R.
Now, for every point P in R and every integer n,
we have
where fm (P) and the remainder Rm (P) are continuous functions of the point P. Moreover,
whenever n > m, as we have assumed that the sequence increases monotonically. In particular, for n > m , the inequality
will hold.
If we consider the sub-sequence of the sequence which
tends to the limit point Q, on account of the continuity
of Rm for fixed values of m., we
also have 
Since
in this limiting process the suffix n increases beyond all
bounds, we may take the index n as large as we please,
because the above inequality holds whenever n > m,
and there are in the sequence of points Pn
tending to Q an infinite number of values of the subscript
n, hence an infinite number of values of n larger
than m. But the relation for all values of m
contradicts the fact that Rm,(Q)
tends to 0 as m increases. Thus, the assumption that the
convergence is non-uniform leads to a contradiction and the theorem
is proved.
1. Show whether the following limits exist:
2. Prove that a function f(x, y) is continuous, if
(a) when y is fixed, f is a continuous function of x, (b) when x is fixed, f is uniformly continuous in y in the sense that there exists for every e a d, independent of x and y, such that
when
3. Prove that f(x, y) is continuous at x = 0, y = 0, if the function F (t,f)=f{tcos f, tsin f) is (a) a continuous function of t when is fixed; (b) uniformly continuous in f when t is fixed, so that there exists for every e a d, independent of t and f, such that
when
4. Prove that the complementary set of a closed set S (i.e., the set of all points not in S) is an open set.
We finally refer to one other special point, the theory of homogeneous functions. The simplest homogeneous functions occurring in analysis and its applications are the homogeneous polynomials in several variables. We say that a function of the form ax+by is a homogeneous function of the first degree in x and y, that a function of the form ax˛+bxy+cy˛ is a homogeneous function of the second degree and, in general, that a polynomial in x and y (or in a larger number of variables) is ahomogeneous function of degree h if in each term the sum of the indices of the independent variables is equal to h, i.e., if the terms (apart from the constant coefficients) are of the form xh, xh-1 y, xh-2y2, ··· , yh. These homogeneous polynomials have the property that the equation
holds for every value of t. We now say, in general, that a function f(x, y, ···)is homogeneous of degree h if it satisfies the equation
Examples of homogeneous functions which are not polynomials are
Another example is the cosine of the angle between two vectors with the respective components x, y, z and u, v, w:
The length of the vector with the components x, y, z
is an example of a function which is positively homogeneous and of the first degree, i.e., the equation defining homogeneous functions does not hold for this function unless t is positive or zero.
Homogeneous functions which are also differentiable satisfy the characteristic Euler relation
In order to prove this statement, we differentiate both sides of the equation f(tx, ty, ···) = thf(x,y) with respect to t; this is permissible, since the equation is an identity in t. Applying the chain rule to the function on the left hand side, we obtain
If we substitute here t = 1, we obtain Euler's formula.
Conversely, it is easy to show that not only is the validity of Euler's relation merely a consequence of the homogeneity of the function f(x, y, . . .), but also the homogeneity of a function is a consequence of Euler's relation, so that Euler's relation is a necessary and sufficient condition for the homogeneity of the function. The fact that a function is homogeneous of degree h can also be expressed by saying that the value of the function divided by xh depends only on the ratios y/x, z/x, ··· . It is therefore sufficient to show that it follows from Euler's relation that, if new variables x=x, h=y/x, z=z/x ··· are introduced, the function
no longer depends on the variable x , i.e., that the equation gx = 0 is an identity. In order to prove this, we use the chain rule:
the expression on the right hand side vanishes by virtue of Euler's relation,and our statement is proved.
This last statement can also be proved in a more elegant but less direct way. We wish to show that there follows from Euler's relation that the function
has the value 0 for all values of t. Obviously, g(l) = 0. Again,
On applying Euler's relation to the arguments tx, ty, ···, we find that
and thus g(t) satisfies the differential equation
If we set g(t) = g(t)th, we obtain g'(t) = h/tg(t) + thg '(t), so that g(t) satisfies the differential equation
which has the unique solution g = const = c. Since obviously for t = 1, one has g(t) = 0, the constant c is 0, whence g(t) = 0 for all values of t, as was to be proved.
1. Prove that, if f(x, y,z, ···) is a homogeneous function of degree h, any k-th derivative of f is a homogeneous function of degree h — k.
2. Prove that for a homogeneous function f of the first degree
