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Boas R.P. — A Primer of Real Functions
Boas R.P. — A Primer of Real Functions



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Название: A Primer of Real Functions

Автор: Boas R.P.

Аннотация:

This is a revised, updated and significantly augmented edition of a classic Carus Monograph (a bestseller for over 25 years) on the theory of functions of a real variable. Earlier editions of this classic covered sets, metric spaces, continuous functions, and differentiable functions. The greatly expanded fourth edition adds sections on measurable sets and functions, the Lebesgue and Stieltjes integrals, and applications.

The book retains the informal, chatty style of the previous editions, remaining accessible to readers with some mathematical sophistication and a background in calculus. The book is thus suitable either for self-study or for supplemental reading in a course on advanced calculus or real analysis.

Not intended as a systematic treatise, this book has more the character of a sequence of lectures on a variety of interesting topics connected with real functions. Many of these topics are not commonly encountered in undergraduate textbooks: for example, the existence of continuous everywhere-oscillating functions (via the Baire category theorem); the universal chord theorem; two functions having equal derivatives, yet not differing by a constant; and application of Stieltjes integration to the speed of convergence of infinite series.

This book recaptures the sense of wonder that was associated with the subject in its early days. A must for your mathematics library.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: Fourth Edition

Год издания: 1996

Количество страниц: 314

Добавлена в каталог: 11.04.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$B_{E}$      110 112
$c_{0}$      23 24 43
$C_{E}$      110 see
$e$      14 120 183 240 264
$f'_{+}$, $f'_{-}$      140
$f(x_{0}^{+})$, $f(x_{0}^{-})$      107
$f^{+}$, $f_{-}$, $f^{+}$, $f_{-}$      139
$f^{-1}$      91
$I^{2}$      24
$L^{2}$      221—224
$L^{2}\cap C$      25 109
$L^{p}$      217
$m$      24 43 44
$m(E)$      197
$R_{n}$      23
$R_{n}$, compact subsets      45 49 51
$R_{n}$, complete      57
$R_{n}$, connected      32 35
$R_{n}$, density in      172—173
$R_{n}$, measure in      73 199
$R_{n}$, separable      43
$R_{n}$, uncountable      15
$\delta$-function      228
$\gamma$      240
$\mu(E)$      172
$\pi$      14 20 237
Abel, Niels Henrik      121 123
Abel, Niels Henrik, convergence test      233
Absolute continuity      204—205
Absolute continuity, of an integral      210 212 216
Absolute convergence      209
Accuracy, numerical      237 242—244
Aczel, Janos      186
Aggregate      2
Agnew, Ralph Palmer      139
Airline space      23
Algebra, fundamental theorem of      59
Algebraic numbers      11—15
Algorithms      21
Almost everywhere      74
Almost everywhere, continuous      206 274
Almost everywhere, convergence      220 221
Almost everywhere, differentiability      155 165—170 204 214 215
Almost everywhere, finite      210
Almost everywhere, limit of continuous functions      202
Almost everywhere, zero derivative      160—164 204 205
Almost uniform convergence      202
Alphabet      3
Alsina, James      122
Alternating Series Test      233
Ampere, Andre Marie      103
Analytic function      47 69 189—192
Antecedent      19
Antiderivative      66 195 206 242
Antipodal points      100—102
antisymmetric      238 279
Apostol, Tom M.      228
Approximation by nonmonotonic functions      70
Approximation by open and closed sets      197—198
Approximation by polygonal functions      128
Approximation by polynomials      128—132
Approximation by rational numbers      75
Approximation by step functions      128
Approximation of a numerical series      237—243
Approximation of an integral      229—230
Approximation of continuous functions      126—132
Approximation of power series by partial sums      47
Arbitrary functions      84—85 87 154—155
Arbitrary shape      101
Arithmetic      7 242 246
Arithmetic mean      181—183
Austin, Donald G.      175
automobile      202
Average      136 181 215 220
Axiom of Choice      xi 200
Aziz, Abdul Kadir      156
B      25 44 47 127
Baire category      see “Category”
Baire class      125 201—202
Baire measure      236
Baire, Rene      125 126
Baire’s theorem      61—72 123—126 192
Baker, Richard L.      xii
Balaguer, Ferran Sunyer      i 72
Bamon, Rodrigo      138
Banach — Tarski paradox      200
Banach, Stefan      72 200
Bartle, Robert G.      205
Base 2 expansion      40—42 115—116 152 161—163
Base 3 expansion      see “Base 2 expansion”
Basis, Hamel      138
Beckenbach, Edwin F.      186
Beekmann, Wolfgang      61
Beesley, E. Maurice      xii 158
Bellman, Richard      186
Benavides, Tomas Dominguez      75
Bernoulli numbers      241—242
Bernoulli polynomials      241
Bernstein equivalence theorem      see “Schroeder — Bernstein theorem”
Bernstein, Felix      21
Bernstein, Sergei      191
Bernstein’s theorem on analytic functions      191—192
Besicovitch, Abram Samoilovitch      73
Bessel’s inequality      222
Bibliography of bibliographies      4
Bicompactness      48
Big game hunting      52
Billions      14 237
Binary expansion      see “Base 2 expansion”
Binomial theorem      191 264
biology      44
Bird trap      46
Bisecting areas and volumes      101—102
Bisection method      46—47
Block of digits      244
Blumberg, H.      90
Boas, Anne Louise      vii
Boas, Harold Philip      vii xii
Boas, Mary Elizabeth Layne      xi
Boas, Ralph Layne      vii
Boas, Ralph Philip      vii 157 192 193 243 244
Boghossian, Artin B.      72
Bolzano — Weierstrass theorem      45—48 50 59 60 91
Bonnet’s theorem      231
Borel, Emile      193
Borel, Emile, measure      236
Borel, Emile, sets      198 272
Borsuk’s theorem      100 101
Borwein, Jonathan M. and Peter B.      21
Bose Majumder, N.C.      138
Botsko, Michael W.      158 210
Bottazzini, Umberto      111
Bound, of derivates      150—151
Bound, positive lower      89
Boundary      27
Boundary, empty      28
Boundary, is closed      30
Boundary, nonempty      35
Boundary, of boundary      28
Boundary, of complement      28
Boundary, of neighborhood      28
Boundary, point      27
Bounded, above      5
Bounded, convergence      111
Bounded, convergence theorem      213
Bounded, derivative      70
Bounded, functional      234
Bounded, functions      25 110—112 209
Bounded, functions, continuous      45 46 48 89
Bounded, functions, not separable      44
Bounded, partial sums      233
Bounded, sequences      24 43 44
Bounded, set      7 26
Bounded, variation      203—204 216 226 228 230 234
Bourbaki, Nicolas      2
Box of maximum volume      182
Brennan, J.G.      104
Briggs, James M.      200
Bromwich, Thomas John l’Anson      123
Brown, Johnny M.      138
Brownian motion      72
Bruckner, Andrew M.      xii 90 155 158
Buck, Ellen F. and Robert Creighton      xi
Bunyakovsky, Victor Ya.      185
Burckel, Robert B.      viii 156 186
C      24 see
C(E)      3
Calculation of $\pi$      21
Calculator      242
Cantor function      161—163 204 216
Cantor function, characterization of      174
Cantor function, integral of      208
Cantor set      39—42 44
Cantor set, arithmetical description      41
Cantor set, distance property      134
Cantor set, first category      64
Cantor set, generalized      74—75 206
Cantor set, irrational points of      42
Cantor set, limit points of      39 40 44
Cantor set, measure zero      74 196
Cantor set, nowhere dense      40
Cantor set, uncountable      41 65
Cantor teepee      42
Cantor’s nested set theorem      62—63
Caratheodory’s criterion      198 200
Cardinality      8—20
Cargo, Gerald T.      xii 157
Carroll, Francis W.      126
Cartan, Henri      193
Cartesian product      79
Catching a lion      46
Category      63—65
Category, for differentiate functions      70—72 193
Category, of level sets      152—153
Category, of points of discontinuity      125—126
Cater, Frank S.      76 174 193
Cauchy sequence      55—58 62 108 110 222 223
Cauchy, Augustin-Louis      108
Cauchy, Augustin-Louis, error of      108 121
Cauchy’s inequality      185 186
Center of gravity      180 193
Cetkovic, S.      157
Chain of elements      18—19
Chain rule      141
Chalice, Donald R.      174
Change of variable      261 268
Characterization of bounded set      7
Characterization of Cantor function      174
Characterization of continuous functions      92
Characterization of linear functional      236
Characterization of polynomials      67—69
Checkerboard      135
Chinn, William G.      104
Chittenden, Edward W.      103
chords      96—100 144 175—180
Chudnovsky, David V. and Gregory V.      21
Civin, Paul      103
Clark, H.M.      xi
class      2
Class, Baire      125 201—202
Clavius      200
clock      100
Closed interval      6 24 28 72
Closed sets      25—37
Closed sets, alternative definitions      29
Closed sets, approximation by      196—198
Closed sets, compact      49
Closed sets, complements of      29
Closed sets, contain limit points      29
Closed sets, intersections of      36—37
Closed sets, level sets      30
Closed sets, measure of      196
Closed sets, nowhere dense      38
Closed sets, of measure zero      174
Closed sets, unions of      36—37
Closure      35—36
Closure, diameter of      61
Closure, of neighborhood      37
Cluster point      29
Cobb, John      44
Coffee      102
Cohen, Paul J.      xi
Collection      1 2
Common error      151
Compact set      48—52
Compact set, closed subset of      49
Compact set, continuous function on      48—49 57 91—93 100 109 112 126—132
Compact set, distance attained for      60—61
Compact set, largest element      59
Competent analyst      186
Complement      3
Complement, boundary of      28
complete      56—58 64 112 222—224
Completion      56
Computation      21 237—243
Computer      20 21 195 203 237 243
Concave      180 181 184
Condensation of singularities      114
Confusion      86 91 108 208 229
Conjecture      160
Connected set      32—35
Connected set, $R_{1}$      35 117
Connected set, $R_{2}$      35
Connected set, continuous function on      92
Connectedness: intrinsic property      34
Consonant      3
Constant function      78
Constant, Euler’s      240
Contain      1 2
continuous curve      114—117
Continuous functions      24—25 83—89
Continuous functions, absolutely      204—205 216
Continuous functions, almost everywhere      206 274
Continuous functions, approximation of      126—132
Continuous functions, at most two-to-one      95—96
Continuous functions, bounded      45 89
Continuous functions, completeness of      112
Continuous functions, convex      175 177
Continuous functions, definitions      85—88
Continuous functions, determined by moments      131—132
Continuous functions, differentiable      140 151
Continuous functions, equivalence of definitions      88—89
Continuous functions, everywhere oscillating      69—70
Continuous functions, fixed points      100
Continuous functions, graphs      85
Continuous functions, intermediate value property      84 92
Continuous functions, limits of      202
Continuous functions, linear      133—134
Continuous functions, linear functionals on      233—236
Continuous functions, maxima      45—46 48—49 92 142
Continuous functions, measurable      201
Continuous functions, neighborhood in space of      26
Continuous functions, nondifferentiable      70—72 153
Continuous functions, of bounded variation      204
Continuous functions, on compact sets      127
Continuous functions, periodic      97—98
Continuous functions, pointwise convergence      109
Continuous functions, pointwise limits      123—126
Continuous functions, positive lower bound      89
Continuous functions, properties of      90—102
1 2 3 4 5
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