# Symmetric Properties of Real Functions 9781003067276, 9780824792305, 9780367402037, 0824792300

This work offers detailed coverage of every important aspect of symmetric structures in function of a single real variab

520 121 3MB

English Pages 472 [470] Year 1994

Cover......Page 1
Half Title......Page 2
Series Page......Page 4
Title Page......Page 8
Preface......Page 10
1.1 Introduction......Page 24
1.2.1 The Even and Odd Parts of a Function......Page 25
1.2.2 Even and Odd Properties of a Function......Page 26
1.2.3 Higher Order Symmetric Differences......Page 27
1.3 Elementary Considerations......Page 28
1.4 Riemann's Theorems......Page 31
1.5 Schwarz Theorem......Page 34
1.6 Auerbach Theorem......Page 36
1.7 Borel Symmetric Derivative......Page 38
1.8 Approximate Symmetric Derivative......Page 40
1.9 Higher Order Symmetric Derivatives......Page 41
1.10 Khintchine Theorem......Page 43
2.1 Introduction......Page 46
2.2 Even and Odd Continuity......Page 47
2.2.1 Stein–Zygmund Theorem......Page 48
2.2.2 Even Version of Stein–Zygmund Theorem......Page 50
2.2.3 A Reduction Theorem......Page 51
2.3.1 Charzyński's Lemma......Page 55
2.3.2 D(f) is Nowhere Dense......Page 56
2.3.3 D(f) is Denumerable......Page 58
2.3.4 D(f) is Scattered......Page 61
2.3.5 The Even Analogue of Charzyriski's Theorem......Page 63
2.5 Wolibner Theorem......Page 64
2.5.1 A Monotonicity Theorem......Page 67
2.6 Jurek–Szpilrajn Example......Page 68
2.7 Pesin–Preiss Theorem......Page 69
2.8 Local Symmetry......Page 71
2.8.1 Points of Local Symmetry......Page 73
2.9 Points of Discontinuity......Page 74
2.9.2 Theorems of Ponomarev and Chlebik......Page 79
2.9.3 Points of Even/Odd Continuity......Page 82
3.1 Introduction......Page 86
3.2 Symmetric Covering Relations......Page 89
3.3 Basic Covering Theorems......Page 91
3.3.2 Charzyński's Covering Theorems......Page 92
3.3.3 Fundamental Covering Theorem for Full Symmetric Covers......Page 94
3.3.4 Other Variants......Page 98
3.3.5 Freiling's Negligent Version......Page 99
3.4 Khintchine Covering Theorem......Page 100
3.5 Uher Covering Lemma......Page 102
3.6 Approximate Symmetric Covering Relations......Page 106
3.6.1 A Lemma for Measurable Covering Relations......Page 107
3.6.2 Analysis of the Covering Properties......Page 108
3.6.3 Partitioning Theorem......Page 111
3.7 2–Interval Partitions......Page 120
3.7.1 Rectangles and 2–Intervals......Page 121
3.7.2 Geometrical Arguments......Page 124
3.7.3 The Covering Theorem......Page 130
3.7.4 Partitions into Squares......Page 132
4.1 Introduction......Page 134
4.2 Midpoint-Linear Functions......Page 135
4.2.2 Hamel Theorem......Page 138
4.2.3 Close to Midpoint-Linear Functions......Page 139
4.2.4 Locally Midpoint-Linear Functions......Page 141
4.3 Midpoint-Convex Functions......Page 142
4.3.1 Theorem of Blumberg–Sierpiński......Page 143
4.3.2 Convex Sets, Anticonvex Sets......Page 145
4.4 Uniform Smoothness Conditions......Page 146
4.4.1 Modulus of Continuity......Page 147
4.4.3 Continuously Differentiable Functions......Page 150
4.4.4 Theorem of M. Weiss and Zygmund......Page 152
4.4.5 Almost Nowhere Differentiable Functions......Page 154
4.4.6 Monotonic, Uniformly Smooth Functions......Page 156
4.4.7 Some Examples......Page 157
4.5 Boundedness......Page 158
4.6 Symmetric Functions......Page 160
4.6.1 Continuity Properties of Symmetric Functions......Page 161
4.6.2 Baire Class......Page 164
4.6.3 Approximately Symmetric Functions......Page 166
4.6.4 Points of Symmetry......Page 169
4.6.5 Points of Approximate Symmetry......Page 170
4.6.7 Determining Sets of Symmetric Functions......Page 171
4.6.8 Derivates of Symmetric Functions......Page 173
4.6.9 Typical Symmetric Functions......Page 174
4.7.1 Continuity Properties......Page 175
4.7.3 Derivates of Quasi-Smooth Functions......Page 176
4.7.4 A Tauberian Theorem......Page 179
4.8.1 Elementary Properties......Page 182
4.8.2 Continuous, Smooth Functions......Page 184
4.8.3 Makarov's Theorems......Page 187
4.8.4 Measurable, Smooth Functions......Page 188
4.8.5 Neugebauer's Example......Page 189
4.8.6 Neugebauer Alternative......Page 190
4.8.7 Approximately Smooth Functions......Page 192
4.8.8 L[sub(p)]–Smooth Functions......Page 193
4.8.9 Discontinuities of Lₚ–Smooth Functions......Page 194
4.8.10 Discontinuities of Approximately Smooth Functions......Page 195
4.9 Super–Smooth Functions......Page 197
4.9.1 Differentiability a.e.......Page 198
4.9.2 Integral of Marcinkiewicz......Page 200
4.9.3 Theorem of Denjoy......Page 201
5.1 Introduction......Page 206
5.2 Some Basic Monotonicity Theorems......Page 207
5.3 Splattered and Scattered Versions......Page 208
5.3.1 Freiling Semi-Scattered Theorem......Page 211
5.3.2 Freiling Scattered Theorem......Page 213
5.3.3 Freiling Proof of Charzyński Theorem......Page 214
5.3.4 The Even Analogue of Charzyiiski Theorem......Page 217
5.4 Evans–Larson Theorem......Page 219
5.5 Mean Value Theorems......Page 221
5.6 Freiling–Rinne Theorem......Page 223
5.7 Convexity Theorems......Page 225
5.7.1 Further Convexity Theorems......Page 228
5.8.1 A Brief History of Symmetric Monotonicity Theorems......Page 233
6.1 Introduction......Page 236
6.2.1 Exact Symmetry......Page 237
6.2.2 Essential Symmetry......Page 241
6.2.3 Exact Local Symmetry Everywhere......Page 245
6.2.4 Points of Local Symmetry of a Set......Page 248
6.2.5 Points of Local Symmetry of a Function......Page 252
6.3 Symmetric Monotonicity......Page 254
6.3.2 Points of Symmetric Increase......Page 255
6.3.3 Porosity Properties......Page 257
6.4 Symmetric Continuity......Page 260
6.4.1 Theorem of Uher......Page 261
6.4.2 Symmetrically Continuous Functions......Page 264
6.4.4 Weak Symmetric Continuity......Page 265
6.6 Boundedness......Page 268
6.7 Symmetric Lipschitz Conditions......Page 270
7.1 Introduction......Page 272
7.2 Extreme Symmetric Derivatives......Page 273
7.2.1 Relations Among the Derivates......Page 274
7.2.3 Nonmeasurable Derivates......Page 278
7.2.4 Baire Class of Symmetric Derivates......Page 279
7.2.6 Porosity Relations for Symmetric Derivates......Page 280
7.2.7 Denjoy Relations for Symmetric Derivates......Page 281
7.3.1 Baire Class of Symmetric Derivatives......Page 285
7.3.2 Symmetric Differentiability......Page 287
7.3.3 Theorem of Belna, Evans and Humke......Page 288
7.3.5 Points of Non Symmetric–Differentiability......Page 289
7.3.6 Infinite Symmetric Derivatives......Page 291
7.3.7 Steep Infinite Derivatives......Page 292
7.3.8 Zero Symmetric Derivative......Page 295
7.3.10 Larson's Primitive......Page 296
7.3.11 The Range of Symmetric Derivatives......Page 299
7.4.1 Baire Class of Approximate Symmetric Derivative......Page 301
7.4.2 Baire Class of Approximate Symmetric Derivates......Page 303
7.4.3 Measurability of the Approximate Symmetric Derivative......Page 306
7.4.4 Approximate Version of Khintchine Theorem......Page 308
7.4.5 Relations for Monotone Functions......Page 309
7.4.6 Typical Continuous Functions......Page 310
7.5 Borel Symmetric Derivative......Page 311
8.1 Introduction......Page 316
8.2 First Order Symmetric Variation......Page 317
8.2.1 Zero Variation......Page 318
8.2.2 Finite Variation......Page 320
8.2.3 Variation and Symmetric Derivates......Page 323
8.3.1 Functions on Rectangles and 2–Intervals......Page 325
8.3.2 Symmetric Increments......Page 327
8.3.3 Variational Definitions......Page 329
8.3.4 Basic Properties......Page 332
8.3.5 Variation of a Continuous Function......Page 336
8.3.6 Zero Variation......Page 337
8.3.7 Differentiation and Variation......Page 339
8.3.8 Finite Variation......Page 343
8.3.9 Generalization of Schwarz Theorem......Page 344
8.5 An Example of Skvorcov......Page 345
9.1 Introduction......Page 346
9.1.1 Integrals from Derivatives......Page 348
9.2.1 Preliminaries......Page 352
9.2.2 A Symmetric Newton Integral......Page 353
9.2.3 An Elementary Application to Trigonometric Series......Page 358
9.2.4 A Symmetric Totalization......Page 359
9.2.5 A Symmetric Variational Integral......Page 364
9.2.6 Symmetric Absolute Continuity......Page 369
9.2.7 A Further Application to Trigonometric Series......Page 374
9.2.8 A Symmetric Perron Integral......Page 375
9.2.9 A Symmetric Riemann Integral......Page 377
9.2.10 Variational Characterization of the (R[sub(s)][sup(1)])–Integral......Page 380
9.2.11 Lusin Type Characterization......Page 384
9.3.1 Preliminaries......Page 385
9.3.2 An Approximate Symmetric Newton Integral......Page 386
9.3.3 An Application to Trigonometric Series......Page 389
9.3.4 An Approximate Symmetric Variational Integral......Page 391
9.3.5 Approximate Symmetric Absolute Continuity......Page 397
9.3.6 An Approximate Symmetric Perron Integral......Page 401
9.3.7 An Approximate Symmetric Riemann Integral......Page 403
9.3.8 Applications to Trigonometric Series......Page 406
9.4 Second Order Symmetric Integrals......Page 411
9.4.1 A Second Order Symmetric Newton Integral......Page 412
9.4.2 The Definitions and an Integrability Criterion......Page 414
9.4.3 Properties of the Integral......Page 417
9.4.4 Integration and Variation......Page 420
9.4.5 An Application to Trigonometric Series......Page 421
9.5 Incompatibilities......Page 422
A.1 Scattered Sets......Page 426
A.2 Scattered Baire Theorem......Page 429
A.3 A Density Computation......Page 431
A.4 Density Points......Page 432
A.5 Category Density Points......Page 433
A.6 Hamel Bases......Page 434
A.7 Weak Quasi–Continuity......Page 435
A.8 Weak Approximate Continuity......Page 436
A.10 Goffman Theorem......Page 437
A.11 Measurability......Page 438
A.12 The Baire Property......Page 440
Problems......Page 444
References......Page 450
Index......Page 464