Counterexamples in analysis:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Mineola, NY
Dover Publ.
2003
|
| Ausgabe: | Unabridged, slightly corr. republ. of the 1965 2. print. of the work orig. publ. in 1964 by Holden-Day, Inc., San Francisco |
| Schriftenreihe: | Dover books on mathematics
|
| Schlagworte: | |
| Online-Zugang: | Klappentext Inhaltsverzeichnis |
| Beschreibung: | Hier auch später erschienene, unveränderte Nachdrucke |
| Beschreibung: | XXIV, 195 S. Ill., graph. Darst. |
| ISBN: | 9780486428758 0486428753 |
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MARC
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| 245 | 1 | 0 | |a Counterexamples in analysis |c Bernard R. Gelbaum ; John M. H. Olmsted |
| 250 | |a Unabridged, slightly corr. republ. of the 1965 2. print. of the work orig. publ. in 1964 by Holden-Day, Inc., San Francisco | ||
| 264 | 1 | |a Mineola, NY |b Dover Publ. |c 2003 | |
| 300 | |a XXIV, 195 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Dover books on mathematics | |
| 500 | |a Hier auch später erschienene, unveränderte Nachdrucke | ||
| 650 | 4 | |a Mathematical analysis | |
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Datensatz im Suchindex
| _version_ | 1804135047102464001 |
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| adam_text | Most mathematical examples illustrate the truth of a statement;
counterexamples demonstrate the falsity of a statement. These
counterexamples, arranged according to difficulty or sophistication,
deal mostly with the part of analysis known as real variables, starting
at the level of calculus.
The first half of the book concerns functions of a real variable; topics
include the real number system, functions and limits, differentiation,
Riemann integration, sequences, infinite series, uniform convergence,
and sets and measure on the real axis. The second half, encompass֊
ing higher dimensions, examines functions of two variables, plane sets,
area, metric and topological spaces, and function spaces.
Each chapter begins with an introduction that determines notation, ter-
minology, and definitions, in addition to offering statements of some
of the more important relevant theorems. This volume contains much
that will prove suitable for students who have not yet completed a first
course in calculus, and ample material of interest to more advanced
students of analysis as well as graduate students.
Dover (200m repubJ¡cation of the edition originally pub-
t ~», ,՝
an
Table of Contents
Part I. Functions of a Real Variable
I. The Real Number System
Introduction 8
1. An infinite field that cannot be ordered 18
2. A field that is an ordered field in two distinct ways 14
3. An ordered field that is not complete 14
4. A non-Archimedean ordered field IS
5. An ordered field that cannot be completed 16
6. An ordered field where the rational numbers are not dense 16
7. An ordered field that is Cauchy-complete but not complete 17
8* An integral domain without unique factorization 17
9. Two numbers without a greatest common divisor 18
10. A fraction that cannot be reduced to lowest terms uniquely 18
II. Functions continuous on a closed interval and failing to
have familiar properties in case the number system is not
complete 18
a. A function continuous on a closed interval and not
bounded there (and therefore, since the interval is
bounded, not uniformly continuous there) 19
b. A function continuous and bounded on a closed interval
but not uniformly continuous there 19
c. A function uniformly continuous (and therefore
bounded) on a closed interval and not possessing a maxi-
mum value there 19
• · ·
Vlll
Table of Contents
d. A function continuous on a closed interval and failing to
have the intermediate value property 19
e. A nonconstant differentiable function whose derivative
vanishes identically over a closed interval 19
f# A differentiable function for which Rollers theorem (and
therefore the law of the mean) fails 19
g. A monotonie uniformly continuous nonconstant function
having the intermediate value property, and whose de-
rivative is identically 0 on an interval 19
2. Functions and Limits
Introduction 20
1. A nowhere continuous function whose absolute value is
everywhere continuous 22
2. A function continuous at one point only (cf. Example 22) 22
3. For an arbitrary noncompact set, a continuous and un-
bounded function having the set as domain 22
4. For an arbitrary noncompact set, an unbounded and locally
bounded function having the set as domain 23
5* A function that is everywhere finite and everywhere locally
unbounded 23
6* For an arbitrary noncompact set, a continuous and bounded
function having the set as domain and assuming no extreme
values 23
7. A bounded function having no relative extrema on a com-
pact domain 24
8. A bounded function that is nowhere semicontinuous 24
9. A nonconstant periodic function without a smallest positive
period 25
10. An irrational function 25
11. A transcendental function 25
12. Functions y — f(u), u£(RJ and u = g(x), a; 6 (ft, whose
composite function y = f(g(x)) is everywhere continuous,
and such that
lim f(u) = c, lim g(x) = 6, lim f(g(x)) ^ c 26
u *b as— a x—* a
ix
Table of Contents
13. Two uniformly continuous functions whose product is not
uniformly continuous 26
14. A function continuous and one-to-one on an interval and
whose inverse is not continuous 27
15. A function continuous at every irrational point and discon-
tinuous at every rational point 27
16. A semieontinuous function with a dense set of points of dis-
continuity 27
17. A function with a dense set of points of discontinuity every
one of which is removable 27
18. A monotonic function whose points of discontinuity form an
arbitrary countable (possibly dense) set 28
19. A function with a dense set of points of continuity, and a
dense set of points of discontinuity no one of which is re-
movable 28
20. A one-to-one correspondence between two intervals that is
nowhere monotonic 28
21. A continuous function that is nowhere monotonic 29
22. A function whose points of discontinuity form an arbitrary
given closed set 80
23. A function whose points of discontinuity form an arbitrary
given Fa set (cf. Example 8, Chapter 4, and Examples 8, 10,
and 22, Chapter 8) 80
24. A function that is not the limit of any sequence of continu-
ous functions (cf. Example 10, Chapter 4) 81
25. A function with domain [0, 1] whose range for every nonde-
generate subinterval of [0, 1] is [0, 1] (cf. Example 27, Chap-
ter 8) 81
26. A discontinuous linear function 88
27. For each n € 91, n(2n + 1) functions j = 1, 2, · · · ,
n, i « 1, 2, * · * , 2n + 1, satisfying:
(а) All are continuous on [0, 1]
(б) For any function ƒ (rci,a:2, * * · , xn) continuous for 0 ^ a?i,
#2, · · * , xn i 1, there are 2n + 1 functions i ~ 1,
2, * · · , 2n + 1, each continuous on (ft, such that
/(xi, X2, · · , Xn) = ¿2 ) SS
X
Table of Contents
3. Differentiation
Introduction 35
1. A function that is not a derivative 35
2. A differentiable function with a discontinuous derivative 36
3. A discontinuous function having everywhere a derivative
(not necessarily finite) 36
4. A differentiable function having an extreme value at a point
where the derivative does not make a simple change in sign 36
5. A differentiable function whose derivative is positive at a
point but which is not monotonic in any neighborhood of
the point 37
6. A function whose derivative is finite but unbounded on a
closed interval 37
7. A function whose derivative exists and is bounded but pos-
sesses no (absolute) extreme values on a closed interval 37
8. A function that is everywhere continuous and nowhere dif-
ferentiable 38
9. A differentiable function for which the law of the mean fails 39
10. An infinitely differentiable function of x that is positive for
positive x and vanishes for negative x /fi
11. An infinitely differentiable function that is positive in the
unit interval and vanishes outside Jfi
12. An infinitely differentiable “bridging function,” equal to 1
. on [1, -f- « ), equal to 0 on (—■« , 0], and strictly monotonic
on [0, 1] 40
13. An infinitely differentiable monotonic function ƒ such that
lim f(x) = 0, lim f (x) ^ 0 40
X —^-f-OO x “ { OO
4. Riemann Integration
Introduction 42
1. A function defined and bounded on a closed interval but not
Riemann-integrable there 42
2. A Riemann-integrable function without a primitive 42
3. A Riemann-integrable function without a primitive on any
interval 43
Table of Contents
4. A function possessing a primitive on a closed interval but
failing to be Riemann-integrable there (cf. Example 35,
Chapter 8) 43
5. A Riemann-integrable function with a dense set of points of
discontinuity 43
6. A function ƒ such that g(x) = j f(t)dt is everywhere dif-
Jo
ferentiable with a derivative different from f(x) on a dense
set 43
7. Two distinct semicontinuous functions at a zero °‘distance” 44
8. A Riemann-integrable function with an arbitrary F* set of
measure zero as its set of points of discontinuity (cf. Ex-
ample 22, Chapter 8) 44
9. A Riemann-integrable function of a Riemann-integrable
function that is not Riemann-integrable (cf. Example 34,
Chapter 8) 44
10. A bounded monotonic limit of Riemann-integrable func-
tions that is not Riemann-integrable (cf. Example 33, Chap-
ter 8) 45
11. A divergent improper integral that possesses a finite Cauchy
principal value 45
12. A convergent improper integral on [1, +od) whose inte-
grand is positive, continuous, and does not approach zero at
infinity 45
13. A convergent improper integral on [0, + 00) whose inte-
grand is unbounded in every interval of the form [a, + co),
where a 0 43
14. Functions ƒ and g such that ƒ is Riemann-Stieltjes integrable
with respect to g on both [a, b] and [b, c], but not on [a, c] 46
5. Sequences
Introduction 47
1. Bounded divergent sequences 47
2. For an arbitrary closed set, a sequence whose set of limit
points is that set 43
3. A divergent sequence {a»} for which lim (an+p — an) = 0
for every positive integer p
• ·
XIX
Table of Contents
4. For an arbitrary strictly increasing sequence { * „}= { K«) 1
of positive integers, a divergent sequence {a,} such that
lim (a^n) — = 0 49
5. Sequences {dn} and {6„} such that ____
lim On + lim bn lim (o„ + bn) lim a„ + lim b„
lim (a„+6„) lim a» + lim bn 50
6. Sequences {«in}, {«2»}, · · · such that
lim (ai» + ö2b ~t՜ ՛ ՛ ) lim oj„ -f- lim ain go
n— H֊ 0 ft—»4*00 n-^-foo
7. Two uniformly convergent sequences of functions the se-
quence of whose products does not converge uniformly 51
8. A divergent sequence of sets 51
9. A sequence {An} of sets that converges to the empty set but
whose cardinal numbers — + 00 51
6. Infinite Series
Introduction 53
1. A divergent series whose general term approaches zero 54
2. A convergent series 2 2 a» and a divergent series X) bn such
that On è bn, n sa 1, 2, · * · 54
3. A convergent series ^2 On and a divergent series ^2 bn such
that lan| è l n|, n « 1, 2, · · · ££
4· For an arbitrary given positive series, either a dominated
divergent series or a dominating convergent series 54
5. A convergent series with a divergent rearrangement 54
6. For an arbitrary conditionally convergent series ^2 an and
an arbitrary real number x} a sequence {¿n}, where en — 1
for n = 1, 2, * · · , such that ^2 enan = x 56
7. Divergent series satisfying any two of the three conditions
of the standard alternating series theorem 56
8. A divergent series whose general term approaches zero and
which, with a suitable introduction of parentheses, becomes
convergent to an arbitrary sum 57
9. For a given positive sequence { *} with limit inferior zero, a
Table of Contents
positive divergent series 23 an whose general term approaches
zero and such that lim an/bn — 0
n—♦֊boo
10. For a given positive sequence {bn} with limit inferior zero, a
positive convergent series 23 a» such that lim an/bn =+00
71
11. For a positive sequence {c„} with limit inferior zero, a posi-
tive convergent series 2] an and a positive divergent series
23 bn such that an/bn — cny n ~ 1,2,···
12. A function positive and continuous for x ^ 1 and such that
r+co
I f(x)dz converges and f(n) diverges
Jl »«1
57
57
58
58
13. A function positive and continuous for x ^ 1 and such that
I f(x)dx diverges and f(n) converges
1
14. Series for which the ratio test fails
15. Series for which the root test fails
16. Series for which the root test succeeds and the ratio test fails
17. Two convergent series whose Cauchy product series di-
verges
18. Two divergent series whose Cauchy product series con-
verges absolutely
19. For a given sequence ^23a™»j , rc =« 1, 2, · · · of positive
4-00
convergent series, a positive convergent series 23 a™ that
772 — 1
does not compare favorably with any series of nj
20. A Toeplitz matrix T and a divergent sequence that is trans-
formed by T into a convergent sequence
21. For a given Toeplitz matrix — a sequence {a,· j where
for each j, aj — ±1, such that the transform { »} of fa,} by
T diverges
22. A power series convergent at only one point (cf. Example
24)
58
59
60
61
61
62
62
64
66
68
XIV
Table of Contents
23. A function whose Maclaurin series converges everywhere
but represents the function at only one point 68
24. A function whose Maclaurin series converges at only one
point 68
25. A convergent trigonometric series that is not a Fourier series 70
26. An infinitely differentiable function f(x) such that
lim ƒ(x) — 0 and that is not the Fourier transform of
| as J *—
any Lebesgue-integrable function 72
27. For an arbitrary countable set C [— x, a continuous
function whose Fourier series diverges at each point of E and
converges at each point of [— x, x]78
28. A (Lebesgue-) integrable function on [— x,x] whose Fourier
series diverges everywhere 73
29. A sequence { an} of rational numbers such that for every
function ƒ continuous on [0, 1] and vanishing at 0 (/(0) = 0)
there exists a strictly increasing sequence { of positive
integers such that, with no = 0:
+oo (
/0*0 = £ s £ a”x” »
j- “0 +1 J
the convergence being uniform on [0, 1] 74
7. Uniform Convergence
Introduction 76
1· A sequence of everywhere discontinuous functions con-
verging uniformly to an everywhere continuous function 76
2. A sequence of infinitely differentiable functions converging
uniformly to zero, the sequence of whose derivatives di-
verges everywhere 76
3. A nonuniform limit of bounded functions that is not
bounded 77
4. A nonuniform limit of continuous functions that is not
continuous 77
5. A nonuniform limit of Riemann-integrable functions that is
not Riemann-integrable (cf. Example 33, Chapter 8) 78
6. A sequence of functions for which the limit of the integrals
is not equal to the integral of the limit 79
XV
Table of Contents
7. A sequence of functions for which the limit of the deriva-
tives is not equal to the derivative of the limit 80
8. Convergence that is uniform on every closed subinterval
but not uniform on the total interval 80
9. A sequence {fn} converging uniformly to zero on [0, + » )
/+CO
fn(x)dx-+* 0 80
10. A series that converges nonuniformly and whose general
term approaches zero uniformly 81
11. A sequence converging nonuniformly and possessing a
uniformly convergent subsequence 81
12. Nonuniformly convergent sequences satisfying any three of
the four conditions of Dini s theorem 81
8. Sets aixd Measure on the Real Axis
Introduction 83
1. A perfect nowhere dense set 85
2. An uncountable set of measure zero 86
3. A set of measure zero whose difference set contains a neigh-
borhood of the origin 87
4. Perfect nowhere dense sets of positive measure 88
5. A perfect nowhere dense set of irrational numbers 89
6. A dense open set whose complement is not of measure zero 90
7* A set of the second category 90
8. A set that is not an F , set 91
9. A set that is not a Gb set 91
10. A set A for which there exists no function having A as its set
of points of discontinuity 91
11. A nonmeasurable set 92
12. A set D such that for every measurable set A,
M*(£ n A) - 0 and m*(Z C A) = v(A) 94
13. A set A of measure zero such that every real number is a
point of condensation of A 94
14. A nowhere dense set A of real numbers and a continuous
mapping of A onto the closed unit interval [0, 1] 95
xvi
Table of Contents
15 A continuous monotonie function with a vanishing deriva-
tive almost everywhere
16. A topological mapping of a closed interval that destroys
both measurability and measure zero
17. A measurable non-Borel set
18. Two continuous functions that do not differ by a constant
but that have everywhere identical derivatives (in the finite
or infinite sense)
19. A set in [0, 1] of measure 1 and category I
20. A set in [0, 1] of measure zero and category II
21. A set of measure zero that is not an F« set
22. A set of measure zero such that there is no function—֊
Riemann-integrable or not — having the set as its set of
points of discontinuity
23. Two perfect nowhere dense sets in [0, 1] that are homeo-
morphic, but exactly one of which has measure zero
24. Two disjoint nonempty nowhere dense sets of real numbers
such that every point of each set is a limit point of the
other
25. Two homeomorphic sets of real numbers that are of differ-
ent category
26. Two homeomorphic sets of real numbers such that one is
dense and the other is nowhere dense
27. A function defined on (ft, equal to zero almost everywhere
and whose range on every nonempty open interval is (ft
28. A function on (ft whose graph is dense in the plane
29. A function ƒ such that 0 S ƒ(#) + °° everywhere but
for every nonempty open interval (a, 6)
30. A continuous strictly monotonic function with a vanishing
derivative almost everywhere
31. A bounded semicontinuous function that is not Riemann-
integrable, nor equivalent to a Riemann-integrable function
32. A bounded measurable function not equivalent to a Rie-
mann-integrable function
96
98
98
98
99
99
100
100
101
101
102
108
104
105
105
105
105
106
XVII
Table of Contents
33· A bounded monotonic limit of continuous functions that is
not Riemann-integrable, nor equivalent to a Riemann-
integrable function (cf. Example 10, Chapter 4)
34· A Riemann-integrable function/, and a continuous function
g, both defined on [0, 1] and such that the composite func-
tion f(g(x)) is not Riemann-integrable on [0, 1], nor equiva-
lent to a Riemann-integrable function there (cf. Example
9, Chapter 4)
35· A bounded function possessing a primitive on a closed inter-
val but failing to be Riemann-integrable there
36· A function whose improper (Riemann) integral exists but
whose Lebesgue integral does not
37. A function that is Lebesgue-measurable but not Borel-
measurable
38· A measurable function fix) and a continuous function g(x)
such that the composite function ƒ (0(2)) is not measurable
39. A continuous monotonic function g(x) and a continuous
function f (x) such that
f f(x)dg(x) 9^ f f(x)g (x)dx
Jo J 0
40. Sequences of functions converging in different senses
41. Two measures ¡i and v on a measure space ( S) such that
/.1 is absolutely continuous with respect to v and for which no
function/exist s such that ju(jE) — / for ail E £S
Part II. Higher Dimensions
9. Functions of Two Variables
Introduction
1. A discontinuous function of two variables that is continuous
in each variable separately
2. A function of two variables possessing no limit at the origin
xviii
106
106
107
108
108
109
109
109
112
115
115
Table of Contents
but for which any straight line approach gives the limit
zero
3. A refinement of the preceding example lie
4. A discontinuous (and hence nondifferentiable) function of
two variables possessing first partial derivatives everywhere 117
5. Functions ƒ for which exactly two of the following exist and
are equal:
lim fix, y), lim lim fix, y), lim lim fix, y) 117
J,) _► ( jt fr) x~֊ a y—+b · y~*b x~ a
6. Functions ƒ for which exactly one of the following exists:
lim fix, y), lim lim fix, y), lim lim fix, y) 117
(x, y)— (a ) x~*a y~+b y-*b x~+a
7. A function ƒ for which
lim lim fix, y) and lim lim f(x, y)
x—*a y—*b y— b x—► «.
exist and are unequal 118
8. A function fix, y) for which lim fix, y) = gix) exists
2/ — 0
uniformly in x, lim fix, y) ֊ ) exists uniformly in y,
x—»0
lim gix) — lim hiy), but lim fix, y) does not exist 118
x-+ 0 V~+ 0 (x, y) — (0, 0)
9. A differentiable function of two variables that is not con-
tinuously differentiable 1;[g
10. A differentiable function with unequal mixed second-order
partial derivatives 120
11. A continuously differentiable function ƒ of two variables x
and y} and a plane region R such that df/dy vanishes iden-
tically in R but ƒ is not independent of y in R 121
12. A locally homogeneous continuously differentiable function
of two variables that is not homogeneous 121
13. A differentiable function of two variables possessing no ex-
tremum at the origin but for which the restriction to an
arbitrary line through the origin has a strict relative mini-
mum there 122
14. A refinement of the preceding example 122
15. A function ƒ for which
È I fix, y) dy * £ [~ fix, y) dy,
XIX
Table of Contents
although each integral is proper
16. A function ƒ for which
f f f(% y) dy dz 9* ( f fix, y) dx dy,
Jq Jq Jo Jo
although each integral is proper
17. A double series ^ amn for which
tn,n
mny
to n
n TO
although convergence holds throughout
18. A differential P dx + Q dy and a plane region R in which
P dx + Q dy is locally exact but not exact
19. A solenoidal vector field defined in a simply-connected re-
gion and possessing no vector potential
10. Plane Sets
Introduction
1. Two disjoint closed sets that are at a zero distance
2. A bounded plane set contained in no minimum closed disk
3. “Thin” connected sets that are not simple arcs
4. Two disjoint plane circuits contained in a square and con-
necting both pairs of opposite vertices
5. A mapping of the interval [0, 1] onto the square [0, 1] X
[0, 1]
6. A space-filling arc in the plane
7. A space-filling arc that is almost everywhere within a count-
able set
8. A space-filling arc that is almost everywhere differentiable
9. A continuous mapping of [0, 1] onto [0, 1] that assumes
every value an uncountable number of times
10. A simple arc in the unit square and of plane measure arbi-
tarily near 1
11. A connected compact set that is not an arc
12. A plane region different from the interior of its closure
13. Three disjoint plane regions with a common frontier
14. A non-Jordan region equal to the interior of its closure
123
124
124
125
126
128
ISO
ISO
131
132
132
1S8
134
134
134
135
138
138
138
139
xx
I
Table of Contents
15 A bounded plane region whose frontier has positive measure
16. A simple arc of infinite length
17՞ A simple arc of infinite length and having a tangent line at
every point
18. A simple are that is of infinite length between every pair of
distinct points on the arc
19. A smooth curve C containing a point P that is never the
nearest point of C to any point on the concave side of C
20. A subset A of the unit square S = [0, 1] X [0, 1] that is
dense in 8 and such that every vertical or horizontal line
that meets S meets A in exactly one point
21. A nonmeasurable plane set having at most two points in
common with any line
22. A nonnegative function of two variables y) such that
140
140
140
141
141
142
142
dx — 0
and such that
ƒ ƒ /0, y) dA, where S = [0, 1] X [0, 1],
s
does not exist 144
23. A real-valued function of one real variable whose graph is a
nonmeasurable plane set 145
24. A connected set that becomes totally disconnected upon the
removal of a single point 146
11. Area
Introduction 11*7
1. A bounded plane set without area 148
2. A compact plane set without area 148
3. A bounded plane region without area 149
4. A bounded plane Jordan region without area 149
5. A simple closed curve whose plane measure is greater than
that of the bounded region that it encloses 149
xxi
Table of Contents
6. Two functions f and p defined on [0, 1] and such that
(a) f (x) ${x) for x 6 [0, 1];
(b) / bf(x) — i {x) dx exists and is equal to 1;
Jo
(c) S ss {(x, y) t (jx) y ՝P(x)} is without area
7. A means of assigning an arbitrarily large finite or infinite
area to the lateral surface of a right circular cylinder
8. For two positive numbers e and M, a surface S in three-
dimensional space such that:
(a) S is homeomorphic to the surface of a sphere;
(b) The surface area of 8 exists and is less than s;
(e) The three-dimensional Lebesgue measure of S exists
and is greater than M
9* A plane set of arbitrarily small plane measure within which
the direction of a line segment of unit length can be reversed
by means of a continuous motion
12. Metric and Topological Spaces
Introduction
1. A decreasing sequence of nonempty closed and bounded sets
with empty intersection
2. An incomplete metric space with the discrete topology
3. A decreasing sequence of nonempty closed balls in a com-
plete metric space with empty intersection
4. Open and closed balls, O and B, respectively, of the same
center and radius and such that B ^ 6.
5. Closed balls Bx and B2j of radii rx and r2 respectively, such
that Bx C B2 and r2
6. A topological space X and a subset Y such that the limit
points of Y do not form a closed set
7. A topological space in which limits of sequences are not
unique
8· A separable space with a nonseparable subspace
9. A separable space not satisfying the second axiom of counta-
bility
xxii
149
150
152
153
154
158
158
158
159
160
160
160
160
161
Table of Contents
10 For a given set, two distinct topologies that have identical
convergent sequences 161
11 A topological space X, a set A C X, and a limit point of A
that is not a limit of any sequence in A 164
12 A topological space X whose points are functions, whose
topology corresponds to pointwise convergence, and which
is not metrizable 166
13. A mapping of one topological space onto another that is
continuous but neither open nor closed 167
14. A mapping of one topological space onto another that is
open and closed but not continuous 167
15. A mapping of one topological space onto another that is
closed but neither continuous nor open 167
16. A mapping of one topological space onto another that is
continuous and open but not closed 168
17. A mapping of one topological space onto another that is
open but neither continuous nor closed 168
18. A mapping of one topological space onto another that is con-
tinuous and closed but not open 169
19. A topological space X, and a subspace Y in which there are
two disjoint open sets not obtainable as intersections of Y
with disjoint open sets of X 169
20. Two nonhomeomorphie topological spaces each of which is
a continuous one-to-one image of the Other 169
21. A decomposition of a three-dimensional Euclidean ball B
into five disjoint subsets Si, S2, Sz, Sa, (where Ss consists
of a single point) and five rigid motions, RXi R2) Rz, R±, Rs
such that
B £* Ri(Si) JRt(S%) S Rz(Sz)^R4(Sa)^JR,(Sb) 170
22. For £, M 0, two Euclidean balls B£ and BM of radius e
and M respectively, a decomposition of Be into a finite num-
ber of disjoint subsets Si, S2, * * · , Sn, and n rigid motions
Ri, R2, · * · , Rn such that
BM - Ri(S1) JR2(S2)yj · * · yjRn(Sn) 171
sexiii
Table of Contents
13. Function Spaces
Introduction 172
1. Two monotonic functions whose sum is not monotonic 175
2. Two periodic functions whose sum is not periodic 175
3. Two semicontinuous functions whose sum is not semicon-
tinuous 175
4. Two functions whose squares are Riemann-integrable and
the square of whose sum is not Riemann-integrable 177
5. Two functions whose squares are Lebesgue-integrable and
the square of whose sum is not Lebesgue-integrable 177
6. A function space that is a linear space but neither an alge-
bra nor a lattice 17s
7. A linear function space that is an algebra but not a lattice 178
8. A linear function space that is a lattice but not an algebra 178
9. Two metrics for the space (7([0? 1]) of functions continuous
on [0, 1] such that the complement of the unit ball in one is
dense in the unit ball of the other 178
Bibliography iso
Special Symbols 188
Index 187
Errata
xxiv
|
| adam_txt |
Most mathematical examples illustrate the truth of a statement;
counterexamples demonstrate the falsity of a statement. These
counterexamples, arranged according to difficulty or sophistication,
deal mostly with the part of analysis known as "real variables," starting
at the level of calculus.
The first half of the book concerns functions of a real variable; topics
include the real number system, functions and limits, differentiation,
Riemann integration, sequences, infinite series, uniform convergence,
and sets and measure on the real axis. The second half, encompass֊
ing higher dimensions, examines functions of two variables, plane sets,
area, metric and topological spaces, and function spaces.
Each chapter begins with an introduction that determines notation, ter-
minology, and definitions, in addition to offering statements of some
of the more important relevant theorems. This volume contains much
that will prove suitable for students who have not yet completed a first
course in calculus, and ample material of interest to more advanced
students of analysis as well as graduate students.
Dover (200m repubJ¡cation of the edition originally pub-
t ~», ,՝
an
Table of Contents
Part I. Functions of a Real Variable
I. The Real Number System
Introduction 8
1. An infinite field that cannot be ordered 18
2. A field that is an ordered field in two distinct ways 14
3. An ordered field that is not complete 14
4. A non-Archimedean ordered field IS
5. An ordered field that cannot be completed 16
6. An ordered field where the rational numbers are not dense 16
7. An ordered field that is Cauchy-complete but not complete 17
8* An integral domain without unique factorization 17
9. Two numbers without a greatest common divisor 18
10. A fraction that cannot be reduced to lowest terms uniquely 18
II. Functions continuous on a closed interval and failing to
have familiar properties in case the number system is not
complete 18
a. A function continuous on a closed interval and not
bounded there (and therefore, since the interval is
bounded, not uniformly continuous there) 19
b. A function continuous and bounded on a closed interval
but not uniformly continuous there 19
c. A function uniformly continuous (and therefore
bounded) on a closed interval and not possessing a maxi-
mum value there 19
• · ·
Vlll
Table of Contents
d. A function continuous on a closed interval and failing to
have the intermediate value property 19
e. A nonconstant differentiable function whose derivative
vanishes identically over a closed interval 19
f# A differentiable function for which Rollers theorem (and
therefore the law of the mean) fails 19
g. A monotonie uniformly continuous nonconstant function
having the intermediate value property, and whose de-
rivative is identically 0 on an interval 19
2. Functions and Limits
Introduction 20
1. A nowhere continuous function whose absolute value is
everywhere continuous 22
2. A function continuous at one point only (cf. Example 22) 22
3. For an arbitrary noncompact set, a continuous and un-
bounded function having the set as domain 22
4. For an arbitrary noncompact set, an unbounded and locally
bounded function having the set as domain 23
5* A function that is everywhere finite and everywhere locally
unbounded 23
6* For an arbitrary noncompact set, a continuous and bounded
function having the set as domain and assuming no extreme
values 23
7. A bounded function having no relative extrema on a com-
pact domain 24
8. A bounded function that is nowhere semicontinuous 24
9. A nonconstant periodic function without a smallest positive
period 25
10. An irrational function 25
11. A transcendental function 25
12. Functions y — f(u), u£(RJ and u = g(x), a; 6 (ft, whose
composite function y = f(g(x)) is everywhere continuous,
and such that
lim f(u) = c, lim g(x) = 6, lim f(g(x)) ^ c 26
u *b as— a x—* a
ix
Table of Contents
13. Two uniformly continuous functions whose product is not
uniformly continuous 26
14. A function continuous and one-to-one on an interval and
whose inverse is not continuous 27
15. A function continuous at every irrational point and discon-
tinuous at every rational point 27
16. A semieontinuous function with a dense set of points of dis-
continuity 27
17. A function with a dense set of points of discontinuity every
one of which is removable 27
18. A monotonic function whose points of discontinuity form an
arbitrary countable (possibly dense) set 28
19. A function with a dense set of points of continuity, and a
dense set of points of discontinuity no one of which is re-
movable 28
20. A one-to-one correspondence between two intervals that is
nowhere monotonic 28
21. A continuous function that is nowhere monotonic 29
22. A function whose points of discontinuity form an arbitrary
given closed set 80
23. A function whose points of discontinuity form an arbitrary
given Fa set (cf. Example 8, Chapter 4, and Examples 8, 10,
and 22, Chapter 8) 80
24. A function that is not the limit of any sequence of continu-
ous functions (cf. Example 10, Chapter 4) 81
25. A function with domain [0, 1] whose range for every nonde-
generate subinterval of [0, 1] is [0, 1] (cf. Example 27, Chap-
ter 8) 81
26. A discontinuous linear function 88
27. For each n € 91, n(2n + 1) functions j = 1, 2, · · · ,
n, i « 1, 2, * · * , 2n + 1, satisfying:
(а) All are continuous on [0, 1]
(б) For any function ƒ (rci,a:2, * * · , xn) continuous for 0 ^ a?i,
#2, · · * , xn i 1, there are 2n + 1 functions i ~ 1,
2, * · · , 2n + 1, each continuous on (ft, such that
/(xi, X2, · ' · , Xn) = ¿2 ) SS
X
Table of Contents
3. Differentiation
Introduction 35
1. A function that is not a derivative 35
2. A differentiable function with a discontinuous derivative 36
3. A discontinuous function having everywhere a derivative
(not necessarily finite) 36
4. A differentiable function having an extreme value at a point
where the derivative does not make a simple change in sign 36
5. A differentiable function whose derivative is positive at a
point but which is not monotonic in any neighborhood of
the point 37
6. A function whose derivative is finite but unbounded on a
closed interval 37
7. A function whose derivative exists and is bounded but pos-
sesses no (absolute) extreme values on a closed interval 37
8. A function that is everywhere continuous and nowhere dif-
ferentiable 38
9. A differentiable function for which the law of the mean fails 39
10. An infinitely differentiable function of x that is positive for
positive x and vanishes for negative x /fi
11. An infinitely differentiable function that is positive in the
unit interval and vanishes outside Jfi
12. An infinitely differentiable “bridging function,” equal to 1
. on [1, -f- « ), equal to 0 on (—■« , 0], and strictly monotonic
on [0, 1] 40
13. An infinitely differentiable monotonic function ƒ such that
lim f(x) = 0, lim f'(x) ^ 0 40
X —^-f-OO x “ { OO
4. Riemann Integration
Introduction 42
1. A function defined and bounded on a closed interval but not
Riemann-integrable there 42
2. A Riemann-integrable function without a primitive 42
3. A Riemann-integrable function without a primitive on any
interval 43
Table of Contents
4. A function possessing a primitive on a closed interval but
failing to be Riemann-integrable there (cf. Example 35,
Chapter 8) 43
5. A Riemann-integrable function with a dense set of points of
discontinuity 43
6. A function ƒ such that g(x) = j f(t)dt is everywhere dif-
Jo
ferentiable with a derivative different from f(x) on a dense
set 43
7. Two distinct semicontinuous functions at a zero °‘distance” 44
8. A Riemann-integrable function with an arbitrary F* set of
measure zero as its set of points of discontinuity (cf. Ex-
ample 22, Chapter 8) 44
9. A Riemann-integrable function of a Riemann-integrable
function that is not Riemann-integrable (cf. Example 34,
Chapter 8) 44
10. A bounded monotonic limit of Riemann-integrable func-
tions that is not Riemann-integrable (cf. Example 33, Chap-
ter 8) 45
11. A divergent improper integral that possesses a finite Cauchy
principal value 45
12. A convergent improper integral on [1, +od) whose inte-
grand is positive, continuous, and does not approach zero at
infinity 45
13. A convergent improper integral on [0, + 00) whose inte-
grand is unbounded in every interval of the form [a, + co),
where a 0 43
14. Functions ƒ and g such that ƒ is Riemann-Stieltjes integrable
with respect to g on both [a, b] and [b, c], but not on [a, c] 46
5. Sequences
Introduction 47
1. Bounded divergent sequences 47
2. For an arbitrary closed set, a sequence whose set of limit
points is that set 43
3. A divergent sequence {a»} for which lim (an+p — an) = 0
for every positive integer p
• ·
XIX
Table of Contents
4. For an arbitrary strictly increasing sequence { * „}= { K«) 1
of positive integers, a divergent sequence {a,} such that
lim (a^n) — = 0 49
5. Sequences {dn} and {6„} such that _
lim On + lim bn lim (o„ + bn) lim a„ + lim b„
lim (a„+6„) lim a» + lim bn 50
6. Sequences {«in}, {«2»}, · · · such that
lim (ai» + ö2b ~t՜ ՛ " ՛ ) lim oj„ -f- lim ain go
n— H֊ 0 ft—»4*00 n-^-foo
7. Two uniformly convergent sequences of functions the se-
quence of whose products does not converge uniformly 51
8. A divergent sequence of sets 51
9. A sequence {An} of sets that converges to the empty set but
whose cardinal numbers — + 00 51
6. Infinite Series
Introduction 53
1. A divergent series whose general term approaches zero 54
2. A convergent series 2 2 a» and a divergent series X) bn such
that On è bn, n sa 1, 2, · * · 54
3. A convergent series ^2 On and a divergent series ^2 bn such
that lan| è l n|, n « 1, 2, · · · ££
4· For an arbitrary given positive series, either a dominated
divergent series or a dominating convergent series 54
5. A convergent series with a divergent rearrangement 54
6. For an arbitrary conditionally convergent series ^2 an and
an arbitrary real number x} a sequence {¿n}, where \en\ — 1
for n = 1, 2, * · · , such that ^2 enan = x 56
7. Divergent series satisfying any two of the three conditions
of the standard alternating series theorem 56
8. A divergent series whose general term approaches zero and
which, with a suitable introduction of parentheses, becomes
convergent to an arbitrary sum 57
9. For a given positive sequence { *} with limit inferior zero, a
Table of Contents
positive divergent series 23 an whose general term approaches
zero and such that lim an/bn — 0
n—♦֊boo
10. For a given positive sequence {bn} with limit inferior zero, a
positive convergent series 23 a» such that lim an/bn =+00
71
11. For a positive sequence {c„} with limit inferior zero, a posi-
tive convergent series 2] an and a positive divergent series
23 bn such that an/bn — cny n ~ 1,2,···
12. A function positive and continuous for x ^ 1 and such that
r+co
I f(x)dz converges and f(n) diverges
Jl »«1
57
57
58
58
13. A function positive and continuous for x ^ 1 and such that
I f(x)dx diverges and f(n) converges
1
14. Series for which the ratio test fails
15. Series for which the root test fails
16. Series for which the root test succeeds and the ratio test fails
17. Two convergent series whose Cauchy product series di-
verges
18. Two divergent series whose Cauchy product series con-
verges absolutely
19. For a given sequence ^23a™»j , rc =« 1, 2, · · · of positive
4-00
convergent series, a positive convergent series 23 a™ that
772 — 1
does not compare favorably with any series of nj
20. A Toeplitz matrix T and a divergent sequence that is trans-
formed by T into a convergent sequence
21. For a given Toeplitz matrix — a sequence {a,· j where
for each j, aj — ±1, such that the transform { »} of fa,} by
T diverges
22. A power series convergent at only one point (cf. Example
24)
58
59
60
61
61
62
62
64
66
68
XIV
Table of Contents
23. A function whose Maclaurin series converges everywhere
but represents the function at only one point 68
24. A function whose Maclaurin series converges at only one
point 68
25. A convergent trigonometric series that is not a Fourier series 70
26. An infinitely differentiable function f(x) such that
lim ƒ(x) — 0 and that is not the Fourier transform of
| as J *—
any Lebesgue-integrable function 72
27. For an arbitrary countable set C [— x, a continuous
function whose Fourier series diverges at each point of E and
converges at each point of [— x, x]78
28. A (Lebesgue-) integrable function on [— x,x] whose Fourier
series diverges everywhere 73
29. A sequence { an} of rational numbers such that for every
function ƒ continuous on [0, 1] and vanishing at 0 (/(0) = 0)
there exists a strictly increasing sequence { of positive
integers such that, with no = 0:
+oo (
/0*0 = £ s £ a”x” »
j-'“0 +1 J
the convergence being uniform on [0, 1] 74
7. Uniform Convergence
Introduction 76
1· A sequence of everywhere discontinuous functions con-
verging uniformly to an everywhere continuous function 76
2. A sequence of infinitely differentiable functions converging
uniformly to zero, the sequence of whose derivatives di-
verges everywhere 76
3. A nonuniform limit of bounded functions that is not
bounded 77
4. A nonuniform limit of continuous functions that is not
continuous 77
5. A nonuniform limit of Riemann-integrable functions that is
not Riemann-integrable (cf. Example 33, Chapter 8) 78
6. A sequence of functions for which the limit of the integrals
is not equal to the integral of the limit 79
XV
Table of Contents
7. A sequence of functions for which the limit of the deriva-
tives is not equal to the derivative of the limit 80
8. Convergence that is uniform on every closed subinterval
but not uniform on the total interval 80
9. A sequence {fn} converging uniformly to zero on [0, + » )
/+CO
fn(x)dx-+* 0 80
10. A series that converges nonuniformly and whose general
term approaches zero uniformly 81
11. A sequence converging nonuniformly and possessing a
uniformly convergent subsequence 81
12. Nonuniformly convergent sequences satisfying any three of
the four conditions of Dini's theorem 81
8. Sets aixd Measure on the Real Axis
Introduction 83
1. A perfect nowhere dense set 85
2. An uncountable set of measure zero 86
3. A set of measure zero whose difference set contains a neigh-
borhood of the origin 87
4. Perfect nowhere dense sets of positive measure 88
5. A perfect nowhere dense set of irrational numbers 89
6. A dense open set whose complement is not of measure zero 90
7* A set of the second category 90
8. A set that is not an F , set 91
9. A set that is not a Gb set 91
10. A set A for which there exists no function having A as its set
of points of discontinuity 91
11. A nonmeasurable set 92
12. A set D such that for every measurable set A,
M*(£ n A) - 0 and m*(Z C\ A) = v(A) 94
13. A set A of measure zero such that every real number is a
point of condensation of A 94
14. A nowhere dense set A of real numbers and a continuous
mapping of A onto the closed unit interval [0, 1] 95
xvi
Table of Contents
15 A continuous monotonie function with a vanishing deriva-
tive almost everywhere
16. A topological mapping of a closed interval that destroys
both measurability and measure zero
17. A measurable non-Borel set
18. Two continuous functions that do not differ by a constant
but that have everywhere identical derivatives (in the finite
or infinite sense)
19. A set in [0, 1] of measure 1 and category I
20. A set in [0, 1] of measure zero and category II
21. A set of measure zero that is not an F« set
22. A set of measure zero such that there is no function—֊
Riemann-integrable or not — having the set as its set of
points of discontinuity
23. Two perfect nowhere dense sets in [0, 1] that are homeo-
morphic, but exactly one of which has measure zero
24. Two disjoint nonempty nowhere dense sets of real numbers
such that every point of each set is a limit point of the
other
25. Two homeomorphic sets of real numbers that are of differ-
ent category
26. Two homeomorphic sets of real numbers such that one is
dense and the other is nowhere dense
27. A function defined on (ft, equal to zero almost everywhere
and whose range on every nonempty open interval is (ft
28. A function on (ft whose graph is dense in the plane
29. A function ƒ such that 0 S ƒ(#) + °° everywhere but
for every nonempty open interval (a, 6)
30. A continuous strictly monotonic function with a vanishing
derivative almost everywhere
31. A bounded semicontinuous function that is not Riemann-
integrable, nor equivalent to a Riemann-integrable function
32. A bounded measurable function not equivalent to a Rie-
mann-integrable function
96
98
98
98
99
99
100
100
101
101
102
108
104
105
105
105
105
106
XVII
Table of Contents
33· A bounded monotonic limit of continuous functions that is
not Riemann-integrable, nor equivalent to a Riemann-
integrable function (cf. Example 10, Chapter 4)
34· A Riemann-integrable function/, and a continuous function
g, both defined on [0, 1] and such that the composite func-
tion f(g(x)) is not Riemann-integrable on [0, 1], nor equiva-
lent to a Riemann-integrable function there (cf. Example
9, Chapter 4)
35· A bounded function possessing a primitive on a closed inter-
val but failing to be Riemann-integrable there
36· A function whose improper (Riemann) integral exists but
whose Lebesgue integral does not
37. A function that is Lebesgue-measurable but not Borel-
measurable
38· A measurable function fix) and a continuous function g(x)
such that the composite function ƒ (0(2)) is not measurable
39. A continuous monotonic function g(x) and a continuous
function f (x) such that
f f(x)dg(x) 9^ f f(x)g'(x)dx
Jo J 0
40. Sequences of functions converging in different senses
41. Two measures ¡i and v on a measure space ( S) such that
/.1 is absolutely continuous with respect to v and for which no
function/exist s such that ju(jE) — / for ail E £S
Part II. Higher Dimensions
9. Functions of Two Variables
Introduction
1. A discontinuous function of two variables that is continuous
in each variable separately
2. A function of two variables possessing no limit at the origin
xviii
106
106
107
108
108
109
109
109
112
115
115
Table of Contents
but for which any straight line approach gives the limit
zero
3. A refinement of the preceding example lie
4. A discontinuous (and hence nondifferentiable) function of
two variables possessing first partial derivatives everywhere 117
5. Functions ƒ for which exactly two of the following exist and
are equal:
lim fix, y), lim lim fix, y), lim lim fix, y) 117
J,) _► ( jt fr) x~֊ a y—+b · y~*b x~ a
6. Functions ƒ for which exactly one of the following exists:
lim fix, y), lim lim fix, y), lim lim fix, y) 117
(x, y)— (a ) x~*a y~+b y-*b x~+a
7. A function ƒ for which
lim lim fix, y) and lim lim f(x, y)
x—*a y—*b y— b x—► «.
exist and are unequal 118
8. A function fix, y) for which lim fix, y) = gix) exists
2/ — 0
uniformly in x, lim fix, y) ֊ ) exists uniformly in y,
x—»0
lim gix) — lim hiy), but lim fix, y) does not exist 118
x-+ 0 V~+ 0 (x, y) — (0, 0)
9. A differentiable function of two variables that is not con-
tinuously differentiable 1;[g
10. A differentiable function with unequal mixed second-order
partial derivatives 120
11. A continuously differentiable function ƒ of two variables x
and y} and a plane region R such that df/dy vanishes iden-
tically in R but ƒ is not independent of y in R 121
12. A locally homogeneous continuously differentiable function
of two variables that is not homogeneous 121
13. A differentiable function of two variables possessing no ex-
tremum at the origin but for which the restriction to an
arbitrary line through the origin has a strict relative mini-
mum there 122
14. A refinement of the preceding example 122
15. A function ƒ for which
È I fix, y) dy * £ [~ fix, y)\dy,
XIX
Table of Contents
although each integral is proper
16. A function ƒ for which
f f f(% y) dy dz 9* ( f fix, y) dx dy,
Jq Jq Jo Jo
although each integral is proper
17. A double series ^ amn for which
tn,n
mny
to n
n TO
although convergence holds throughout
18. A differential P dx + Q dy and a plane region R in which
P dx + Q dy is locally exact but not exact
19. A solenoidal vector field defined in a simply-connected re-
gion and possessing no vector potential
10. Plane Sets
Introduction
1. Two disjoint closed sets that are at a zero distance
2. A bounded plane set contained in no minimum closed disk
3. “Thin” connected sets that are not simple arcs
4. Two disjoint plane circuits contained in a square and con-
necting both pairs of opposite vertices
5. A mapping of the interval [0, 1] onto the square [0, 1] X
[0, 1]
6. A space-filling arc in the plane
7. A space-filling arc that is almost everywhere within a count-
able set
8. A space-filling arc that is almost everywhere differentiable
9. A continuous mapping of [0, 1] onto [0, 1] that assumes
every value an uncountable number of times
10. A simple arc in the unit square and of plane measure arbi-
tarily near 1
11. A connected compact set that is not an arc
12. A plane region different from the interior of its closure
13. Three disjoint plane regions with a common frontier
14. A non-Jordan region equal to the interior of its closure
123
124
124
125
126
128
ISO
ISO
131
132
132
1S8
134
134
134
135
138
138
138
139
xx
I
Table of Contents
15 A bounded plane region whose frontier has positive measure
16. A simple arc of infinite length
17՞ A simple arc of infinite length and having a tangent line at
every point
18. A simple are that is of infinite length between every pair of
distinct points on the arc
19. A smooth curve C containing a point P that is never the
nearest point of C to any point on the concave side of C
20. A subset A of the unit square S = [0, 1] X [0, 1] that is
dense in 8 and such that every vertical or horizontal line
that meets S meets A in exactly one point
21. A nonmeasurable plane set having at most two points in
common with any line
22. A nonnegative function of two variables y) such that
140
140
140
141
141
142
142
dx — 0
and such that
ƒ ƒ /0, y) dA, where S = [0, 1] X [0, 1],
s
does not exist 144
23. A real-valued function of one real variable whose graph is a
nonmeasurable plane set 145
24. A connected set that becomes totally disconnected upon the
removal of a single point 146
11. Area
Introduction 11*7
1. A bounded plane set without area 148
2. A compact plane set without area 148
3. A bounded plane region without area 149
4. A bounded plane Jordan region without area 149
5. A simple closed curve whose plane measure is greater than
that of the bounded region that it encloses 149
xxi
Table of Contents
6. Two functions f and \p defined on [0, 1] and such that
(a) f (x) ${x) for x 6 [0, 1];
(b) / bf(x) — i {x)\dx exists and is equal to 1;
Jo
(c) S ss {(x, y) \ t (jx) y ՝P(x)} is without area
7. A means of assigning an arbitrarily large finite or infinite
area to the lateral surface of a right circular cylinder
8. For two positive numbers e and M, a surface S in three-
dimensional space such that:
(a) S is homeomorphic to the surface of a sphere;
(b) The surface area of 8 exists and is less than s;
(e) The three-dimensional Lebesgue measure of S exists
and is greater than M
9* A plane set of arbitrarily small plane measure within which
the direction of a line segment of unit length can be reversed
by means of a continuous motion
12. Metric and Topological Spaces
Introduction
1. A decreasing sequence of nonempty closed and bounded sets
with empty intersection
2. An incomplete metric space with the discrete topology
3. A decreasing sequence of nonempty closed balls in a com-
plete metric space with empty intersection
4. Open and closed balls, O and B, respectively, of the same
center and radius and such that B ^ 6.
5. Closed balls Bx and B2j of radii rx and r2 respectively, such
that Bx C B2 and r2
6. A topological space X and a subset Y such that the limit
points of Y do not form a closed set
7. A topological space in which limits of sequences are not
unique
8· A separable space with a nonseparable subspace
9. A separable space not satisfying the second axiom of counta-
bility
xxii
149
150
152
153
154
158
158
158
159
160
160
160
160
161
Table of Contents
10 For a given set, two distinct topologies that have identical
convergent sequences 161
11 A topological space X, a set A C X, and a limit point of A
that is not a limit of any sequence in A 164
12 A topological space X whose points are functions, whose
topology corresponds to pointwise convergence, and which
is not metrizable 166
13. A mapping of one topological space onto another that is
continuous but neither open nor closed 167
14. A mapping of one topological space onto another that is
open and closed but not continuous 167
15. A mapping of one topological space onto another that is
closed but neither continuous nor open 167
16. A mapping of one topological space onto another that is
continuous and open but not closed 168
17. A mapping of one topological space onto another that is
open but neither continuous nor closed 168
18. A mapping of one topological space onto another that is con-
tinuous and closed but not open 169
19. A topological space X, and a subspace Y in which there are
two disjoint open sets not obtainable as intersections of Y
with disjoint open sets of X 169
20. Two nonhomeomorphie topological spaces each of which is
a continuous one-to-one image of the Other 169
21. A decomposition of a three-dimensional Euclidean ball B
into five disjoint subsets Si, S2, Sz, Sa, (where Ss consists
of a single point) and five rigid motions, RXi R2) Rz, R±, Rs
such that
B £* Ri(Si)\JRt(S%) S Rz(Sz)^R4(Sa)^JR,(Sb) 170
22. For £, M 0, two Euclidean balls B£ and BM of radius e
and M respectively, a decomposition of Be into a finite num-
ber of disjoint subsets Si, S2, * * · , Sn, and n rigid motions
Ri, R2, · * · , Rn such that
BM - Ri(S1)\JR2(S2)yj · * · yjRn(Sn) 171
sexiii
Table of Contents
13. Function Spaces
Introduction 172
1. Two monotonic functions whose sum is not monotonic 175
2. Two periodic functions whose sum is not periodic 175
3. Two semicontinuous functions whose sum is not semicon-
tinuous 175
4. Two functions whose squares are Riemann-integrable and
the square of whose sum is not Riemann-integrable 177
5. Two functions whose squares are Lebesgue-integrable and
the square of whose sum is not Lebesgue-integrable 177
6. A function space that is a linear space but neither an alge-
bra nor a lattice 17s
7. A linear function space that is an algebra but not a lattice 178
8. A linear function space that is a lattice but not an algebra 178
9. Two metrics for the space (7([0? 1]) of functions continuous
on [0, 1] such that the complement of the unit ball in one is
dense in the unit ball of the other 178
Bibliography iso
Special Symbols 188
Index 187
Errata
xxiv |
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| dewey-ones | 515 - Analysis |
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| discipline | Mathematik |
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| edition | Unabridged, slightly corr. republ. of the 1965 2. print. of the work orig. publ. in 1964 by Holden-Day, Inc., San Francisco |
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| index_date | 2024-07-02T13:43:46Z |
| indexdate | 2024-07-09T20:34:17Z |
| institution | BVB |
| isbn | 9780486428758 0486428753 |
| language | English |
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| spelling | Gelbaum, Bernard R. 1922-2005 Verfasser (DE-588)135764076 aut Counterexamples in analysis Bernard R. Gelbaum ; John M. H. Olmsted Unabridged, slightly corr. republ. of the 1965 2. print. of the work orig. publ. in 1964 by Holden-Day, Inc., San Francisco Mineola, NY Dover Publ. 2003 XXIV, 195 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Dover books on mathematics Hier auch später erschienene, unveränderte Nachdrucke Mathematical analysis Beweis (DE-588)4132532-1 gnd rswk-swf Analysis (DE-588)4001865-9 gnd rswk-swf (DE-588)4144384-6 Beispielsammlung gnd-content Analysis (DE-588)4001865-9 s Beweis (DE-588)4132532-1 s DE-604 Olmsted, John Meigs Hubbell 1911-1997 Verfasser (DE-588)135764254 aut Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014589476&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Klappentext Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014589476&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gelbaum, Bernard R. 1922-2005 Olmsted, John Meigs Hubbell 1911-1997 Counterexamples in analysis Mathematical analysis Beweis (DE-588)4132532-1 gnd Analysis (DE-588)4001865-9 gnd |
| subject_GND | (DE-588)4132532-1 (DE-588)4001865-9 (DE-588)4144384-6 |
| title | Counterexamples in analysis |
| title_auth | Counterexamples in analysis |
| title_exact_search | Counterexamples in analysis |
| title_exact_search_txtP | Counterexamples in analysis |
| title_full | Counterexamples in analysis Bernard R. Gelbaum ; John M. H. Olmsted |
| title_fullStr | Counterexamples in analysis Bernard R. Gelbaum ; John M. H. Olmsted |
| title_full_unstemmed | Counterexamples in analysis Bernard R. Gelbaum ; John M. H. Olmsted |
| title_short | Counterexamples in analysis |
| title_sort | counterexamples in analysis |
| topic | Mathematical analysis Beweis (DE-588)4132532-1 gnd Analysis (DE-588)4001865-9 gnd |
| topic_facet | Mathematical analysis Beweis Analysis Beispielsammlung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014589476&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014589476&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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