titles in this series · 2019. 2. 12. · titles in this series volume 5 andrew bruckner...

Titles in This Series Volume 5 Andre w Bruckner

Differentiation o f real functions 1994

4 Davi d Ruelle Dynamical zeta functions for piecewise monotone maps of the interval 1994

3 V . Kumar Murty Introduction to Abelian varieties 1993

2 M . Ya. Antimirov, A. A. Kolyshkin, and Remi Vaillancourt Applied integral transforms 1993

1 D . V. Voiculescu, K. J. Dykema, and A. Nica Free random variables 1992

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Volume 5

CRM MONOGRAPH SERIES Centre d e Recherches Mathematique s Universite d e Montrea l

Differentiation o f Real Function s

Andrew Bruckne r

The Centr e d e Recherche s Mathematique s (CRM ) o f th e Universite d e Montrea l wa s create d i n 196 8 t o promot e research i n pur e an d applie d mathematic s an d relate d disciplines. Amon g it s activitie s ar e specia l them e years , summer schools , workshops , postdoctora l programs , an d publishing. Th e CR M i s supporte d b y th e Universit e d e Montreal, th e Provinc e o f Quebe c (FCAR) , an d th e Natural Science s an d Engineerin g Researc h Counci l o f Canada. I t i s affiliate d wit h th e Institu t de s Science s Mathematiques (ISM ) o f Montreal , whos e constituen t members ar e Concordi a University , McGil l University , th e Universite d e Montreal , th e Universit e d u Quebe c a Montreal, an d th e Ecol e Polytechnique .

^ t t E M ^ .

American Mathematical Society Providence, Rhode Island US A

https://doi.org/10.1090/crmm/005

T h e produc t io n o f this volum e wa s suppor te d i n pa r t b y th e Fond s pou r l a Format io n de Chercheur s e t l 'Aid e a l a Recherch e (Fond s F C A R ) an d th e Na tu ra l Science s an d Engineering Researc h Counci l o f C a n a d a (NSERC) .

1991 Mathematics Subject Classification. Primar y 26A24 ; Secondary 26A21 , 26A27 , 26A48 .

Library o f Congres s Cataloging-in-Publicat io n D a t a

Bruckner, Andre w M . Differentiation o f rea l functions/Andre w Bruckner.—[2n d ed. ]

p. cm.—(CR M monograp h series ; v . 5 ) Includes bibliographica l reference s an d indexes . ISBN 0-8218-6990- 6 1. Calculus , Differential . 2 . Function s o f rea l variables . I . Title . II . Series .

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Table o f Content s

Preface t o th e Secon d Editio n i x

Preface x i

Introduction 1

Premilinaries 3

Chapter 1 . Darbou x Function s 5

1. Example s o f Darboux Function s 5 2. Remark s 5 3. Darbou x function s an d continuit y 6 4. Operations , combinations , an d approximation s 6 5. Additiona l remark s 7

Chapter 2 . Darbou x Function s i n the Firs t Clas s o f Baire 9 1. Equivalence s 9 2. Example s 1 1 3. Operations , combination s an d approximation s 1 2 4. Th e clas s o f derivatives : preliminar y comparison s wit h VBi 1 4 5. Approximat e continuit y 1 5 6. Th e Luzin-Menchof f Theore m an d construction s o f approximatel y

continuous function s 2 0 7. MaximofF s Theorem s 2 6 8. Integra l comparison s o f C, Cap, A' , an d VB\ 2 6 9. Remark s 3 1

Chapter 3 . Continuit y an d Approximat e Continuit y o f Derivatives 3 3 1. Example s o f discontinuous derivative s 3 3 2. Characterizatio n o f the se t o f discontinuitie s o f a derivative 3 4 3. Approximat e continuit y o f the derivativ e 3 5 4. A relationship betwee n C ap an d A ' 3 6

Chapter 4 . Th e Extrem e Derivate s o f a Functio n 3 9 1. Definition s an d basi c propertie s 3 9 2. Measurabilit y an d Bair e classification s o f extreme derivate s 4 0 3. A Darboux-like propert y o f Din i derivative s 4 3 4. Relationship s amon g th e derivate s 4 5

vi TABL E O F CONTENT S

Chapter 5 . Reconstructio n o f the Primitiv e 5 1 1. Reconstruction s b y Rieman n o r Lebesgu e integratio n 5 1 2. Reconstructio n o f the primitiv e whe n it s derivativ e i s finite 5 2 3. Ambiguitie s whe n derivative s ca n b e infinit e 5 6 4. Generalize d bounde d variatio n an d generalize d absolut e continuit y 5 7

Chapter 6 . Th e Zahorsk i Classe s 6 1 1. Definition s an d basi c propertie s 6 1 2. Derivative s an d th e classe s 6 2 3. Relate d condition s 6 6

Chapter 7 . Th e Proble m o f Characterizin g Derivative s 6 9 1. Associate d set s 7 0 2. Perfec t system s 7 0 3. A n analogu e t o characterizin g integral s 7 2 4. A characterization o f A ' 7 2 5. Miscellaneou s remark s 7 5

Chapter 8 . Derivative s a.e . an d Generalization s 7 7 1. Derivative s a.e . 7 7 2. A generalized derivativ e 7 9 3. Universa l generalize d antiderivative s 8 1 4. Differentiabilit y a.e . 8 3

Chapter 9 . Transformation s vi a Homeomorphism s 8 5 1. Differentiabilit y vi a inne r homeomorphism s 8 5 2. Differentiabilit y vi a oute r homeomorphism s 9 0 3. Derivative s vi a inne r homeomorphism s 9 2 4. Derivative s vi a oute r homeomorphism s 9 5 5. Summar y an d miscellaneou s remark s 9 8

Chapter 10 . Generalize d Derivative s 10 1 1. Th e approximat e derivative—basi c propertie s 10 1 2. Behavio r o f approximate derivative s 10 3 3. Miscellan y 11 0 4. Othe r generalize d derivative s 11 2

Chapter 11 . Monotonicit y 11 9 1. Som e historica l backgroun d fo r Sectio n 2 11 9 2. A genera l theore m 12 1 3. Application s o f Theorem 2. 5 12 5 4. Monotonicit y condition s i n term s o f extreme derivate s 12 8 5. Monotonicit y whe n D+F € # i 13 0 6. Convexit y 13 1

Chapter 12 . Stationar y an d Determinin g Set s 13 5 1. Th e stationar y an d determinin g set s fo r certai n classe s 13 5 2. Miscellaneou s remark s 13 9

TABLE O F CONTENT S vi i

Chapter 13 . Behavio r o f Typica l Continuou s Function s 14 1 1. Preliminarie s an d basi c terminolog y 14 1 2. Differentiabilit y structur e o f typica l continuou s function s 14 2 3. Horizonta l leve l sets 14 4 4. Tota l leve l set structur e 14 6 5. Miscellaneou s comment s 14 8

Chapter 14 . Miscellaneou s Topic s 15 1 1. Restrictiv e differentiabilit y propertie s o f functions 15 1 2. Extension s t o derivative s 15 2 3. Th e se t o f points o f differentiability o f a functio n 15 4 4. Derivatives , approximat e continuity , an d summabilit y 15 5 5. Additiona l topic s 15 6

Chapter 15 . Recen t Development s 15 7 1. Pat h derivative s 15 7 2. Th e algebr a generate d b y A ' 16 2 3. Mor e abou t typica l behavio r 16 5 4. Miscellan y 16 9

Bibliography 18 1

Supplementary Bibliograph y 18 9

Terminology Inde x 19 3

Notational Inde x

Preface t o th e Secon d Editio n

Since th e appearanc e o f th e firs t editio n o f thi s boo k i n 1978 , substantia l ad -vances hav e bee n achieve d i n a numbe r o f area s relate d t o th e differentiatio n o f real functions . I t seeme d appropriat e t o ad d a chapte r t o summarize som e of thes e advances. W e hav e attempte d t o suppl y suc h a summar y i n Chapte r 15 . W e limi t this summar y t o ne w wor k tha t extends , o r i s otherwis e closel y connecte d to , th e material tha t appeare d i n th e origina l edition . Eve n here , w e cit e onl y a fractio n of the man y releven t ne w works .

Where convenient , w e provide a n outline o f a new development , stat e ne w the -orems i n detai l an d provid e example s an d proofs . Bu t ofte n th e require d technica l machinery to do this is prohibitive fo r our purpose. I n such cases, we limit ourselve s to indication s o f developments o r results .

The origina l fourtee n chapter s remai n essentiall y unchanged . W e correcte d typographical errors , an d mad e othe r mino r changes .

The manuscrip t wa s abl y type d b y Caroly n Johnson , Phylli s Claudi o an d m y wife Judy , wh o als o helpe d considerabl y wit h th e writin g o f Chapte r 15 . Man y mathematicians spotte d numerou s typographica l errors . Thi s was particularly tru e of Ja n Mafi k an d Ti m Steele , whos e shar p eye s caugh t man y error s tha t woul d otherwise hav e slippe d b y unnoticed . T o al l thes e people , I a m grateful .

Santa Barbar a August, 199 3

Preface

It ha s no w bee n abou t fort y year s sinc e th e publicatio n o f Saks ' book , Theory of the Integral, a book which deals considerably wit h topics which are related to dif-ferentiation theory . Sinc e that time , particularly sinc e the publication o f Zahorski' s paper [216 ] in 1950 , much work has been don e related t o th e differentiation o f rea l functions, bu t littl e o f i t ha s appeare d i n book form .

A consequenc e o f this i s that man y result s hav e bee n reprove d o r rediscovere d several times. I n addition, there ar e many instances of an author proving a theore m "from scratch" , tha t is , withou t th e knowledg e o f relate d result s whic h existe d a t the time , an d whic h could have been used t o prove the theorem much more simply . It therefor e seem s desirabl e t o hav e a boo k whic h

1. provide s a relativel y efficien t developmen t o f th e presen t stat e o f knowl -edge o n th e subject ,

2. discusse s som e o f the ope n problem s whic h ar e wort h investigatin g , an d 3. provide s reference s t o wor k o n topic s whic h th e boo k doesn' t develo p i n

detail. These are the main purposes of the present Notes . I t i s an outgrowth o f courses

and seminar s whic h w e hav e give n from time-to-tim e durin g th e las t fiftee n year s at th e Universit y o f California , Sant a Barbara .

In orde r t o kee p thi s wor k t o manageabl e proportions , w e hav e ha d t o mak e certain compromises . W e ten d t o omi t proof s o f thos e theorem s whic h ar e eithe r readily accessibl e i n standar d books , o r whic h ar e periphera l t o ou r work . O n occasion, whe n severa l theorem s hav e simila r proofs , w e prov e onl y on e o r tw o of thes e theorems . Wher e w e d o no t giv e a complet e proof , however , w e provid e references.

In puttin g thi s boo k together , w e hav e benefite d fro m discussion s wit h man y students an d colleagues . Particula r thank s ar e du e t o Stev e Agronsky , Rober t Biskner an d Donal d Hancoc k wh o hav e carefull y rea d th e entir e manuscrip t an d made helpfu l suggestions . W e als o wis h t o than k Ms . Soni a Ospin a wh o type d the manuscrip t quickl y an d efficiently , an d deal t wit h th e whim s o f th e autho r cheerfully.

Bibliography

1. S . J . Agronsky , A generalization of a theorem of Maximoff and applications, Trans . Amer . Math. Soc . 27 3 (1982) , no . 2 , 767-779 .

2. B . Babcock , On properties of the approximate Peano derivatives, Trans . Amer . Math . Soc . 212 (1975) , 279-294 .

3. A . Baernstei n an d D . Waterman , Functions whose Fourier series converge uniformly for every change of variable, Indian a Univ . Math . J . 2 2 (1972) , 569-576 .

4. A . P . Baisnab , On a theorem of Goffman and Neugebauer, Proc . Amer . Math . Soc . 2 3 (1969), 573-579 .

5. S . Banach , Uber die Baire'sche Kategorie gewisser Funftionenmengen, Studi a Math . 3 (1931), 174-179 .

6. A . Besicovitch , Diskussion der stetigen Funktionen im Zusammenhang mit der Frage uber ihre Differentierbarkeit, Bull . Acad . Sci . d e Russi e 1 9 (1925) , 527-540 .

7. H . Blumberg , New properties of all real functions, Trans . Amer . Math . Soc . 2 4 (1922) , 113-128.

8. N . Bobo c an d S . Marcus , Sur la determination d'une fonction par les valeurs prises sur un certain ensemble, Ann . Sci . Ecol e Norm . Sup . 7 6 (1959) , no . 3 , 151-159 .

9. K . Bogel , Uber die mehrdimensionale Differentiation, Jber . Deutsch . Math . Ver . 6 5 (1962) , 45-71.

10. J . Brown , Lusin density and Ceder's differentiate restrictions of arbitrary real functions, Fund. Math . L X X X I V (1974) , 35-45 .

11. A . Bruckner , On derivatives with a dense set of zeros, Rev . Math . Pure s Appl . 1 0 (1965) , 149-153.

12. , Stationary sets for certain classes of derivates of Darboux functions, Mich . Math . J . 11 (1964) , 305-309 .

13. , A theorem on monotonicity and a solution to a problem of Zahorski, Bull . Amer . Math. Soc . 7 1 (1965) , 713-716 .

14. , An affirmative answer to a problem of Zahorski and some consequences, Mich . Math . J. 1 3 (1966) , 15-26 .

15. , On characterizing classes of functions in terms of associated sets, Canad . Math . Bull. 1 0 (1967) , 227-232 .

16. , Some remarks on extreme derivates, Canad . Math . Bull . 1 2 (1969) , 385-388 . 17. , On the differentiation of integrals in euclidean spaces, Fund . Math . L X V I (1969) ,

129-135. 18. , Density preserving homeomorphisms and a theorem of Maximoff, Quart . J . Math .

Oxford 2 1 (1970) , 337-347 . 19. , Differentiation of integrals, Amer . Math . Monthl y 7 8 (1971) , no . 9 Par t II , 1-51 . 20. , The differentiability properties of typical functions in C[a,b], Amer . Math . Monthl y

80 (1973) , 679-683 . 21. , A theorem on extending derivatives, Boll . U . M . I . 1 0 (1974) , 490-499 . 22. , On transformations of derivatives, Proc . Amer . Math . Soc . 4 8 (1975) , 101-107 . 23. , Inflexible derivatives, Quart . J . Math . Oxfor d 2 9 (1978) , 1-10 . 24. , Differentiability a.e. and approximate differentiability a.e., Proc . Amer . Math . Soc .

66 (1977) , 294-298 . 25. , A general convexity criterion, Glasnik . Math . 13(33 ) (1978) , 231-235 .

181

182 BIBLIOGRAPHY

26. , Some observations about Denjoy's preponderant derivative, Bull . Math . Soc . Sci . Math. R . S . R . 21(69 ) (1977) , 3-10 .

27. A . Bruckne r an d J . Bruckner , Darboux transformations, Trans . Amer . Math . Soc . 12 8 (1967), 103-111 .

28. A . Bruckne r an d J . Ceder , Darboux continuity, Jber . Deutsch . Math . Ver . 6 7 (1965) , 9 3 -117.

29. , On the sum of Darboux functions, Proc . Amer . Math . Soc . 5 1 (1975) , 97-102 . 30. A . Bruckner , J . Cede r an d R . Keston , Representations and approximations by Darboux

functions in the first class of Baire, Rev . Roum . Math . Pure s Appl . 1 3 (1968) , 1247-1254 . 31. A . Bruckner , J . Cede r an d M . Weiss , On the differentiability structure of real functions,

Trans. Amer . Math . Soc . 14 2 (1969) , 1-13 . 32. , On uniform limits of Darboux functions, Colloq . Math . X V (1966) , 65-77 . 33. A . Bruckner , R . Davie s an d C . Goffman , Transformations into Baire 1 functions, Proc .

Amer. Math . Soc . 6 7 (1977) , 62-66 . 34. A . Bruckne r an d K . Garg , The level structure of a residual set of continuous functions,

Trans. Amer . Math . Soc . 23 2 (1977) , 307-321 . 35. A . Bruckne r an d C . Goffman , Differentiability through change of variables, Proc . Amer .

Math. Soc . 6 1 (1976) , 235-241 . 36. , The boundary behavior of real functions in the upper half plane, Rev . Roum . Math .

Pures Appl . X I (1966) , 507-518 . 37. A . Bruckne r an d J . Leonard , On differentiate functions having an everywhere dense set

of intervals of constancy, Canad . Math . Bull . 8 (1965) , 73-76 . 38. , Stationary sets and determining sets for certain classes of Darboux functions, Proc .

Amer. Math . Soc . 1 6 (1965) , 935-940 . 39. , Derivatives, Amer . Math . Monthl y 73 , Par t I I (1966) , 24-56 . 40. A . Bruckne r an d M . Rosenfeld , A theorem on approximate directional derivatives, Ann .

Scuola Norm . Sup . Pis a X X I I (1968) , 343-350 . 41. , On topologizing measure spaces via differentiation bases, Ann . Scuol a Norm . Sup .

Pisa X X I I I (1969) , 243-258 . 42. A . Bruckne r an d C . Thorne , Density covers, Ann . Scuol a Norm . Sup . Pis a X X V I I (1973) ,

97-104. 43. P . Bulle n an d S . Mukhopadhyay , On the Peano derivatives, Canad . J . Math . X X V (1973) ,

127-140. 44. P . Butze r an d H . Berens , Semi-groups of Operators and Approximation, di e Grundlehre n

der Mathematische n Wissenschaften , vol . 145 , Springer , 1967 . 45. J . Ceder , On maximally resolvable spaces, Fund . Math . 5 5 (1964) , 87-93 . 46. , On representing functions by Darboux functions, Act a Sci . Math . Szege d X X V I

(1965), 283-288 . 47. , Differentiate roads for real functions, Fund . Math . 6 5 (1969) , 351-358 . 48. G . Choquet , Application des proprietes descriptives de la fonction contingent a la theorie

des fonctions de variable reelle et a la geometrie differentielle des varietes cartesiennes, J . Math. Pure s Appl . 2 6 (1947) , no . 9 , 115-226 .

49. H . Croft , A note on a Darboux continuous function, J . Lond . Math . Soc . 3 8 (1963) , 9-10 . 50. A . Csaszar , Sur une generalization de la notion de derivee, Act a Sci . Math . Szege d 1 6

(1955), 137-159 . 51. G . Darboux , Memoire sur les fonctions discontinues, Ann . Sci . Scuola Norm. Sup . 4 (1875) ,

57-112. 52. A . Denjoy , Memoire sur la totalisation des nombres derives non-sommables, Ann . Ecol e

Norm. Sup . 3 3 (1916) , 127-236 ; 3 4 (1917) , 181-238 . 53. , Memoire sur les nombres derives des fonctions continues, J . Math . Pure s e t Appl .

1 (1915) , no . 7 , 105-240 . 54. , Sur les fonctions derivees sommables, Bull . Soc . Math . Franc e 4 3 (1915) , 161-248 . 55. , Sur une propriete des fonctions derivees, Enseignemen t Math . 1 8 (1916) , 320-328 . 56. U . Dini , Fondamenti per la Teorica delle Funzioni di Variabili Rreali, Pisa , 1878 . 57. S . Eilenber g an d S . Saks , Sur la derivation des fonctions dans des ensembles denombrables,

Fund. Math . 2 5 (1935) , 264-266 . 58. M . Evans , L p derivatives and approximate Peano derivatives, Trans . Amer . Math . Soc . 16 5

(1972), 381-388 .

BIBLIOGRAPHY 183

59. H . Fast , Une remarque sur la propriete de Weierstrass, Colloq . Math . 7 (1959) , 75-77 . 60. F . Filipczak , Sur les derivees symetriques des fonctions approximativement continues, Col -

loq. Math . 3 4 (1975-76) , 249-256 . 61. , Sur les fonctions continues relativement monotones, Fund . Math . 5 8 (1966) , 75-87 . 62. R . Fleissner , Multiplication and the Fundamental Theorem of Calculus: a survey, Rea l

Anal. Ex . 2 (1976) , 7-34 . 63. , On the products of derivatives, Fund . Math . 8 8 (1975) , 173-178 . 64. R . Fleissne r an d J . Foran , Correspondenc e t o autho r (1976) . 65. , Transformations of differentiable functions, Colloq . Math . 3 9 (1978) , 277-284 . 66. J . Foran , The symmetric and ordinary derivatives, Rea l Anal . Ex . 2 (1977) , no . 2 , 105-108 . 67. , On the product of derivatives, Fund . Math . 8 0 (1973) , 293-294 . 68. M . Frechet , Sur diverses definitions de la differentiabilite, Enseignemen t Math . 1 0 (1964) ,

no. 2 , 177-228 . 69. K . Garg , An analogue of Denjoy's theorem, Ganit a 1 2 (1961) , 9-14 . 70. , Applications of Denjoy analog I (Sufficient conditions for a function to be mono-

tone), Ann . Polon . Math . 1 5 (1964) , 159-165 . 71. , Applications of Denjoy analog II (Local structure of level sets and Dini derivates),

Acta Math . Acad . Sci . Hung . 1 4 (1963) , 183-186 . 72. , Applications of Denjoy analog III (Distribution of various typical level sets), Act a

Math. Acad . Sci . Hung . 1 4 (1963) , 187-195 . 73. , On nowhere monotone functions, HI (Functions of the first and second species),

Rev. Math . Pure s Appl . 8 (1963) , 83-90 . 74. , On bilateral derivates and the derivative, Trans . Amer . Math . Soc . 21 0 (1975) ,

295-329. 75. , On a new definition of derivative, Bull . Amer . Math . Soc . 8 3 (1976) . 76. , Some new notions of derivative, Memoir s Amer . Math . Soc . (t o appear) . 77. , Extensions of a theorem of Denjoy to bilateral derivates (t o appear) . 78. J . Gillis , Note on a conjecture of Erdos, Quart . J . Math . Oxfor d 1 0 (1939) , 151-154 . 79. A . Gleyzal , Interval functions, Duk e Math . J . 8 (1941) , 223-230 . 80. C . Goffman , Real Functions, Rinehart , Ne w York , 1960 . 81. , On Lebesgue's density theorem, Proc . Amer . Math . Soc . 1 (1950) , 384-388 . 82. C . Goffma n an d C . Neugebauer , Linear continuous functions, Proc . Amer . Math . Soc . 1 2

(1961), 997-998 . 83. , On approximate derivatives, Proc . Amer . Math . Soc . 1 1 (1960) , 962-966 . 84. C . Goffman , C . Neugebauer , an d T . Nishiura , Density topology and approximate continuity,

Duke Math . J 2 8 (1961) , 497-506 . 85. C . Goffma n an d D . Waterman , A characterization of the class of functions whose Fourier

series converge for every change of variable, J . Lond . Math . Soc . 1 0 (1975) , 69-74 . 86. W . Gorman , The homeomorphic transformation of c-sets into d-sets, Proc . Amer . Math .

Soc. 1 7 (1966) , 825-830 . 87. , Lebesgue equivalence to functions of the first Baire class, Proc . Amer . Math . Soc .

17 (1966) , 831-834 . 88. M . d e Guzman , Differentiation of integrals in R n , Lectur e Note s i n Mathematics , vol . 481,

Springer, 1975 . 89. O . Hajek , Note sur la mesurabilite B de la derivee superieure, Fund . Math . 4 4 (1957) ,

238-240. 90. P . Halmos , Measure Theory, D . Va n Nostran d Co. , 1950 . 91. C . Haye s an d C . Pauc , Derivation and Martingales, Springer , Berlin , 1970 . 92. C . E . Hobson , Theory of Functions of a Real Variable, vol . 1 , Dover , Ne w York , 1957 . 93. M . Iosifescu , Conditions that the product of two derivatives be a derivative, Rev . Math .

Pures Appl . 4 (1959) , 641-649 . (Russian ) 94. V . Jarnfk , Uber die Differenzierbarkeit stetiger Funktionen, Fund . Math . 2 1 (1933) , 48-58 . 95. , Sur les nombres derivees approximatifs, Fund . Math . 2 2 (1934) , 4-16 . 96. , Sur la derivabilite des fonctions continues, Spis y Privodov . Fak . Univ . Karlov y 12 9

(1934), 3-9 . 97. R . Jeffery , The theory of functions of a real variable, Mathemtica l Exposition s No . 6 ,

University o f Toront o Press , 1962 . 98. E . Kampke , Das Lesbesgue-Stieltjes-Integral, Leipzig , 1956 .

184 BIBLIOGRAPHY

99. E . Kempisty , Sur les fonctions quasicontinues, Fund . Math . 1 9 (1932) , 184-197 . 100. H . Kenyo n an d A . P . Morse , Web Derivatives, Memoir s Amer . Math . Soc . 13 2 (1973) . 101. A . Khintchine , Recherches sur la structure des fonctions mesurables, Fund . Math . 9 (1927) ,

212-279. 102. D . Kolzow , Differentiation von Massen, Lectur e Note s i n Mathematics , vol . 64 , Springer ,

Berlin, 1968 . 103. P . Kostyrko , On the symmetric derivative, Colloq . Math . 2 5 (1972) , 265-267 . 104. Y . Kuboto , A characterization of the approximately continuous Denjoy integral, Canad . J .

Math. X X I I (1970) , 219-226 . 105. M . Kulbacka , Sur certain proprieties des derivees approximatives, Bull . Acad . Polon . Sci .

Ser. Sci . Math . Astronom . Phys . 1 2 (1964) , 17-20 . 106. , Sur les ensembles stationaires et determinants pour certaines classes des derivees

symetriques, Colloq . Math . X I X (1968) , 255-259 . 107. N . Kundu , On uniform symmetric derivatives, Colloq . Math . 2 4 (1972) , 259-269 . 108. , On approximate symmetric derivatives, Colloq . Math . 2 8 (1973) , 275-285 . 109. , Symmetric derivatives and approximate symmetric derivatives of continuous func-

tions, Rev . Roum . Math . Pure s Appl . 1 8 (1973) , 689-697 . 110. , Denjoy theorem for symmetric derivatives, Rev . Roum. Math . Pure s Appl . 1 8 (1973) ,

699-700. 111. , On symmetric derivatives and on properties of Zahorski, Czeck . Math . J 2 6 (101 )

(1976), 154-160 . 112. C . Kuratowski , Topologie I, 4t h ed. , Monographi e Matematyczne , vol . 20 , Warszawa, 1958 . 113. C . Kuratowsk i an d W . Sierpinski , Les fonctions de classe 1 et les ensembles connexes

punctiformes, Fund . Math . 3 (1922) , 303-313 . 114. M . Laczkovich , Separation properties of some subclasses of Baire 1 functions, Act a Math .

Acad. Sci . Hung . 2 6 (1975) , 405-412 . 115. , Separation of sets by derivatives, Act a Math . Acad . Sci . Hung . 2 7 (1976) , 201-207 . 116. , Separation of sets by bounded derivatives, Period . Math . Hung . 7 (1976) , 169-177 . 117. , On the Baire class of selective derivatives, Act a Math . Acad . Sci . Hung . 2 9 (1977) ,

99-105. 118. M . Laczkovic h an d G . Petruska , On the transformers of derivatives, Fund . Math . 10 0

(1978), 179-199 . 119. C . Lee , On functions with summable approximate Peano derivatives, Proc . Amer . Math .

Soc. 5 7 (1976) , 53-57 . 120. , An approximate extension of Cesaro-Perron integrals., Bull . Inst . Math . Acad . Sin .

4 (1976) , 73-82 . 121. , On the approximate Peano derivative, J . Lond . Math . Soc . 1 2 (1976) , 475-478 . 122. C . Le e an d R . O'Malley , The second approximate derivative and the second approximate

Peano derivative, Bull . Inst . Math . Acad . Sin . 3 (1975) , 193-197 . 123. J . Leonard , Some conditions implying the monotonicity of a real function, Rev . Roum .

Math. Pure s Appl . 1 2 (1972) , 757-780 . 124. A . Lindenbaum , Ann . Soc . Polon . Math . 6 (1927) , 129 . 125. J . Lipinski , Mesure et derivee, Colloq . Math . 8 (1961) , 83-88 . 126. , Sur les derivees de Pompeiu, Rev . Math . Pure s Appl . 1 0 (1965) , 447-451 . 127. , Sur certains problemes de Choquet et de Zahorski concernant les fonctions derivees,

Fund. Math . 4 4 (1957) , 94-102 . 128. , Sur la discontinuity approximative et la derivee approximative, Colloq . Math . 1 0

(1963), 103-109 . 129. , Sur les ensembles {f'{x) > a}, Fund . Math . 4 5 (1958) , 254-260 . 130. , Sur les fonctions approximativement continues, Colloq . Math . 5 (1958) , 172-175 . 131. , line propriete des ensembles {f'(x) > a}, Fund . Math . 4 2 (1955) , 339-342 . 132. N . Manna , On Dini derivatives and on a property of Denjoy, Monatsh . Math . 7 4 (1970) ,

323-329. 133. J . Marcinkiewicz , Sur les nombres derives, Fund . Math . 2 4 (1935) , 305-308 . 134. , Sur les series de Fourier, Fund . Math . 2 7 (1937) , 38-69 . 135. J . Marcinkiewic z an d A . Zygmond , On the differentiability of functions and summability

of trigonometric series, Fund . Math . 2 6 (1936) , 1-43 .

BIBLIOGRAPHY 185

136. S . Marcus , Les ensembles stationnaires de certaines classes de fonctions, C . R . Acad . Sci . Paris 25 4 (1962) , 1186-1188 .

137. , Les ensembles stationnaires de certaines classes de fonctions derivees, Att i Accad . Naz. Lince i Rend . CI . Sci . Fis . Mat . Nat . 3 2 (1962) , 484-487 .

138. , On a theorem of Den joy and on approximate derivatives, Monatsh . Math . 6 6 (1962) , 435-440.

139. , Remarques sur les fonctions integrables au sens de Riemann, Bull . Math . Soc . Sci . Math. Phys . R . P . R . 2(50 ) (1958) , no . 4 , 433-439 .

140. , Sur la derivee approximative qualitative, Com . Acad . R . P . R . 3 (1953) , 361-364 . 141. , Sur la limite approximative qualitative, Com . Acad . R . P . R . 3 (1953) , 9-12 . 142. , Sur les derivees dont les zeros forment un ensemble frontiere partout dense, Rend .

Circ. Mat . Palerm o 1 2 (1963) , no . 2 , 1-36 . 143. , Sur les ensembles determinants des derivees approximatives, C . R . Acad . Sci . Pari s

255 (1962) , 1685-1687 . 144. , Sur les ensembles stationnaires des fonctions derivees -finies ou infinies, Com . Acad .

R. P . R . 1 2 (1962) , 399-402 ; (Romanian ; Russia n an d Frenc h summaries) . 145. , Sur une generalization de la notion de quasi-analyticite, C . R . Acad . Sci . Pari s 25 4

(1962), 985-987 . 146. N . Martin , A note on metric density of sets of real numbers, Proc . Amer . Math . Soc . 1 1

(1960), 344-347 . 147. J . Massera , Sobre las fonciones derivables, Bol . Fac . Ingen. Agrimens . Montevide o 2 (1944) ,

647-668. 148. I . Maximoff , Sur les fonctions ayant la propriete de Darboux, Proce . Matem . Fizyc . 4 3

(1936), 241-265 . 149. , Sur les fonctions derivees, Bull . Sci . Math . 6 4 (1940) , 116-121 . 150. , On approximately (p n, A n) continuous functions, Tohok u Math . J . 4 8 (1941) , 34-42 . 151. , On continuous transformation of some functions into an ordinary derivative, Ann .

Scuola Norm . Sup . Pis a 1 2 (1943) , 147-160 . 152. , Sur la transformation continue des fonctions, Bull . Soc . Phys . Math . Kaza n 1 2

(1940), no . 3 , 9-41 ; (Russian ; Frenc h summary) . 153. , Sur la transformation continue des quelques fonctions en derivees exactes, Bull . Soc .

Phys. Math . Kaza n 1 2 (1940) , no . 3 , 57-81 ; (Russian ; Frenc h summary) . 154. S . Mazurkiewicz , Sur les fonctions non-derivables, Studi a Math . 3 (1931) , 92-94 . 155. L . Misik , Bemerkungen uber Approximative Ableitung, Math . Cas . 1 9 (1969) , 283-292 . 156. , Uber Approximative derivierte Zahlen monotoner Funktionen, Czeck . Math . J 2 6

(101) (1976) , 579-583 . 157. A . Morse , A continuous function with no unilateral derivatives, Trans . Amer . Math . Soc .

44 (1938) , 496-507 . 158. , Dini derivatives of continuous functions, Proc . Amer . Math . Soc . 5 (1954) , 126-130 . 159. S . Mukhopadhyay, On Schwarz differentiability IV, Act a Math . Acad . Sci . Hung. 1 7 (1966) ,

129. 160. , On Schwarz differentiability, Proc . Nat . Acad . Sc . Ind . (1966) . 161. _ , On approximate Schwarz differentiability, Monatsh . Math . 7 0 (1966) , 454-460 . 162. , On Dini derivatives and on certain classes of Zahorski, Colloq . Math . 2 5 (1972) ,

105-111. 163. S . Mukhopadhya y an d B . Lahiri , On Schwarz derivatives of continuous functions, Rev .

Roum. Math . Pure s Appl . 1 2 (1967) , 861-864 . 164. I . Natanson , Theory of Functions of a Real Variable, vol . 1 , Ungar , Ne w York , 1961 . 165. , Theory of Functions of a Real Variable, vol . 2 , Ungar , Ne w York , undated . 166. C . Neugebauer , A class of functions determined by dense sets, Arch . Math . 1 2 (1961) ,

206-209. 167. , Darboux functions of Baire class 1 and derivatives, Proc . Amer . Math . Soc . 1 3

(1962), 838-843 . 168. , Differentiability almost everywhere, Proc . Amer . Math . Soc . 1 6 (1965) , 1205-1210 . 169. , A theorem on derivatives, Act a Sci . Math . Szege d 2 3 (1962) , 79-81 . 170. H . Oliver , The exact Peano derivative, Trans . Amer . Math . Soc . 7 6 (1954) , 444-456 . 171. R . O'Malley , Selective Derivates, Act a Math . Acad . Sci . Hung . 2 9 (1977) , 77-97 .

186 BIBLIOGRAPHY

172. , Approximately differentiate functions: the r-topology, Pacifi c J . Math . 7 3 (1977) , 30-46.

173. , M 3 functions, Indian a Univ . Math . J . 2 4 (1974) , 585-591 . 174. , The set where an approximate derivative is a derivative, Proc . Amer . Math . Soc . 5 4

(1976), 122-124 . 175. R . O'Malle y an d C . Weil , The oscillatory behavior of certain derivatives, Trans . Amer .

Math. Soc . 23 4 (1977) , 467-481 . 176. G . Petruska , On the structure of derivatives, Studi a Sci . Math . Hung . 9 (1974) , 251-266 . 177. , Separation and approximation theorems on derivatives, Act a Math . Acad . Sci . Hung .

25 (1974) , 435-442 . 178. G . Petrusk a an d M . Laczkovich , A theorem on approximately continuous functions, Act a

Math. Acad . Sci . Hung . 2 4 (1974) , 383-387 . 179. , Baire 1 functions, approximately continuous functions, and derivatives, Act a Math .

Acad. Sci . Hung . 2 5 (1974) , 189-212 . 180. B . Pettineo, Quelques observations sur les fonctions derivables presque partout, C . R . Acad .

Sci. Pari s 24 8 (1959) , 518-520 . 181. D . Pompeiu , Sur les fonctions derivees, Math . Ann . 6 3 (1906) , 326-332 . 182. D . Preiss , Limits of derivatives and Darboux-Baire functions, Rev . Roum . Math . Pure s

Appl. 1 4 (1969) , 1201-1206 . 183. , On the first derivative of real functions, Comm . Math . Univ . Carol . 1 1 (1974) ,

817-822. 184. , Approximate derivatives and Baire classes, Czeck . Math . J 2 1 (1971) , 373-382 . 185. H . Pu an d J . Chen , Note on approximate continuity and derivates, Inst . Math . Acad . Sinic a

(1974), 1-29 . 186. R . Rinehar t an d J . Wilson , Two types of differentiability of functions on algebras, Rend .

Circ. Mat . Palerm o 1 1 (1962) , no . 2 , 204-216 . 187. A . Rosenthal, On differentiation of integrals and approximate continuity, Bull . Amer. Math .

Soc. 4 8 (1942) , 414-420 . 188. S . Saks , On the functions of Besicovitch in the space of continuous functions, Fund . Math .

19 (1932) , 211-219 . 189. , Sur les nombres derives des fonctions, Fund . Math . 5 (1924) , 98-104 . 190. , Theory of the Integral, Monographi e Matematyczne , vol . 7 , Warszaw-Lwow , 1937 . 191. R . Salem , Essais sur les series trigonometriques, Act a Sci . Ind. , vol . 862 , Hermann , Paris ,

1940. 192. J . Scholz , Essential derivatives of functions in C[a,b] (t o appear) . 193. H . Sen , Darboux's property and its applications, Proc . Benare s Mat . Sco . (N.S. ) 2 (1940) ,

17-23. 194. W . Sierpinski , Sur une propriete des fonctions quelconques d'une variable reelle, Fund .

Math. 2 5 (1935) , 1-4 . 195. , Sur la decomposition des espaces metriques en emsembles disjoints, Fund . Math . 3 6

(1949), 68-71 . 196. G . Sindalovski , Derivatives of continuous functions, Izv . Akad . Nau k SSS R 3 2 (1968) .

(Russian) 197. A . Singh , On infinite derivatives, Fund . Math . 3 3 (1945) , 106-107 .

198a. F . Sunye r Balaguer , Sobre la determinacion de una funcion mediante sus numeros deriva-dos, Collect . Math . 1 0 (1958) , 185-194 .

198b. , Sur la determination d'une fonction par ses nombres derives, C . R . Acad . Sci . Pari s 245 (1957) , 1690-1692 .

199. T . Swiatkowski , On the conditions of monotonicity of functions, Fund . Math . 5 9 (1966) , 189-201.

200. H . Thielman , Theory of Functions of a Real Variable, Prentice-Hall , Englewoo d Cliffs , N . J., 1953 .

201. G . Tolstoff , Sur la derivee approximative exacte, Rec . Math . (Mat . Sbornik ) N . S . (1938) , 499-504.

202. , Sur quelques proprietes des fonctions approximativement continues, Rec . Math . (Mat. Sbornik ) N.-S . 5 (1939) , 637-645 .

203. , Parametricheskoe differentsirovanie i Uzkii Integral Denjoy, Mat . Sborni k 5 3 (1961) , 387-392.

BIBLIOGRAPHY 187

204. Verblunsky , On the Peano derivatives, Proc . Lond . Math . Soc . 3 3 (1971) , 313-324 . 205. V . Volterra , Sui principii del calcolo integrate, Giorn . d i Battaglin i 1 9 (1881) , 333-372 . 206. C . Weil , On properties of derivatives, Trans . Amer . Math . Soc . 11 4 (1965) , 363-376 . 207. , On approximate and Peano derivatives, Proc . Amer . Math . Soc . 2 0 (1969) , 487-490 . 208. , A property for certain derivatives, Indian a Univ . Math . J . 2 3 (1973-74) , 527-536 . 209. , On nowhere monotone functions, Proc . Amer . Math . Soc . 5 6 (1976) , 388-389 . 210. , Monotonicity, convexity and symmetric derivates, Trans . Amer . Math . Soc . 22 2

(1976), 225-237 . 211. H . Whitney , On totally differentiable and smooth functions, Pacifi c J . Math . 1 (1951) ,

143-159. 212. W . Wilkosz , Some properties of derivative functions, Fund . Math . 2 (1921) , 145-154 . 213. G . Young, On the derivates of a function, Proc . Lond. Math . Soc . 1 5 (1916) , no. 2 , 360-384 . 214. W . Young , A theorem in the theory of functions of a real variable, Rend . Circ . Mat. Palerm o

24 (1907) , 187-192 . 215. , A note on the property of being a differential coefficient, Proc . Lond . Math . Soc . 9

(1911), 360-368 . 216. Z . Zahorski , Sur la primiere derivee, Trans . Amer . Math . Soc . 6 9 (1950) , 1-54 . 217. Z . Zalcwasser , Sur les fonctions de Kopcke, Prac e Mat . Fiz . 3 5 (1927-28) , 57-99 . (Polish :

French summary ) 218. A . Zygmund , Trigonometric Series, vol . 1 , Universit y Press , Cambridge , 1959 .

Supplementary Bibliograph y

219. S . Agronsky , A . Bruckner , M . Laczkovic h an d D . Preiss , Convexity conditions and intersec-tions with smooth functions, Trans . Amer . Math . Soc . 28 9 (1985) , 659-677 .

220. S . Agronsky , R . Biskner , A . Bruckne r an d J . Mafik , Representations of functions by deriva-tives, Trans . Amer . Math . Soc . 26 3 (1981) , 493-500 .

221. A . Alikhani-Koopaei , Borel measurability of extreme path derivatives, Rea l Anal . Ex . 1 2 (1986-87), no . 1 , 216-246 .

222. , Extreme path derivatives of a function with respect to a nonporous system of paths, Real Anal . Ex . 1 4 (1988-89) , no . 1 , 77-98 .

223. C . Belna , G . Cargo , M . Evans an d P . Humke, Analogues to the Denjoy-Young-Saks Theorem, Trans. Amer . Math . Soc . 27 1 (1982) , 253-260 .

224. J . Bobok , On nondifferentiable measure-preserving functions, Rea l Anal . Ex . 1 6 (1990-91) , no. 1 , 119-129 .

225. J . Brown , Continuous-, derivative-, and differentiability- restrictions of measurable func-tions, Fund . Math . 14 1 (1992) , 85-95 .

226. A . Bruckner , On destruction of derivatives via changes of scale, Act a Math . Sinic a 9 (1981) , no. 3 , 407-415 .

227. A . Bruckne r an d J . Haussermann , Strong porosity features of typical continuous functions, Acta Math . Hung . 4 5 (1985) , 7-13 .

228. A . Bruckne r an d K . Johnson , Path derivatives and growth control, Proc . Amer . Math . Soc . 91 (1984) , 46-48 .

229. A . Bruckner , M . Laczkovich, G . Petruska an d B . Thomson, Porosity and approximate deriva-tives, Canad . J . Math . 3 8 (1986) , 1149-1180 .

230. A . Bruckner , J . Mafi k an d C . Weil , Some aspects of products of derivatives, Amer . Math . Monthly 9 9 (1992) , no . 2 , 134-145 .

231. , Baire one null functions, Contemp . Math . 4 2 (1985) , 29-41 . 232. A . Bruckner , R . O'Malle y an d B . Thomson , Path Derivatives: A unified view of certain

generalized derivatives, Trans . Amer . Math . Soc . 28 3 (1984) , 97-125 . 233. A . Bruckner an d G . Petruska , Some typical results on bounded Baire 1 functions, Act a Math .

Hung. 4 3 (1984) , 325-333 . 234. A . Bruckne r an d B . Thomson , Porosity estimates for the Dini derivatives, Rea l Anal . Ex . 9

(1983-84), no . 2 , 508-538 . 235. Z . Buczolich , For every continuous f there is an absolutely continuous g such that { / = g}

is not bilaterally strongly porous, Proc . Amer . Math . Soc . 10 0 (1987) , 485-488 . 236. , Sets of convexity of continuous functions, Act a Math . Hung . 5 2 (1988) , 291-303 . 237. P . Bullen , Denjoy's index and porosity, Rea l Anal . Ex . 1 0 (1984-85) , no . 1 , 85-144 . 238. F . Cater , A vector space of Morse functions, Rea l Anal . Ex . 1 3 (1987-88) , no . 1 , 373-389 . 239. J . Cede r an d T . Pearson , Most continuous functions are weird, Period . Math . Hung . 1 2

(1981), 294-304 . 240. J . Cede r an d G . Petruska , Most Darboux Baire 1 functions map big sets onto small sets,

Acta Math . Hung . 4 1 (1983) , 37-46 . 241. F . Clarke , Optimization and Nonsmooth Analysis, Classic s i n Applie d Mathematics , vol . 5 ,

Society fo r Industria l an d Applie d Mathematics , Philadelphia , 1990 . 242. F . Clarke , R . Ster n an d P . Wolenski , Subgradient criteria for monotonicity and the Lipschitz

condition, Canad . J . Math , (t o appear) . 243. M . Codyks , On the set of points at which the derivative equals -j-oc and — oo, Mat . Sborni k

43(85) (1957) , 429-450 . (Russian )

189

190 SUPPLEMENTARY BIBLIOGRAPH Y

244. C . Cordy , On the relationship between the external intersection condition and the intersec-tion condition for path derivatives, Rea l Anal . Ex . 1 3 (1987-88) , no . 2 , 420-431 .

245. M . Evan s an d P . Humke , The equality of unilateral derivates, Proc . Amer . Math . Soc . 7 9 (1980), 609-613 .

246. , The approximate symmetric behavior of real functions, Rea l Anal . Ex . 1 4 (1988-89) , no. 2 , 375-392 .

247. M . Evan s an d C . Weil , Peano derivatives: a survey, Rea l Anal . Ex . 7 (1981-82) , no . 1 , 5-24 . 248. , The approximate symmetric behavior of real functions, Rea l Anal . Ex . 1 4 (1988-89) ,

no. 2 , 375-392 . 249. H . Fejzic , Decompositions of Peano derivatives, Proc . Amer . Math . Soc . (t o appear) . 250. , On approximate Peano derivatives, Act a Math . Hung , (t o appear) . 251. , On generalized Peano and Peano derivatives, Fund . Math , (t o appear) . 252. H . Fejzic,J . Mafi k an d C . Weil , Extending Peano derivatives (t o appear) . 253. C . Freilin g an d D . Rinne , A symmetric density property, monotonicity and the approximate

symmetric derivative, Proc . Amer . Math . Soc . 10 4 (1988) , 1098-1102 . 254. C . Freilin g an d D . Rinne , A symmetric density property for measurable sets, Rea l Anal . Ex .

14 (1988-89) , no . 1 , 203-209 . 255. P . Gandin i an d A . Zucco , Porosity and typical properties of real-valued continuous functions,

Abh. Math . Sem . U . Hambur g 5 9 (1989) , 15-22 . 256. D . Hancock , Derivatives and Lebesgue points via homeomorphic changes of scale, Trans .

Amer. Math . Soc . 26 7 (1981) , no . 1 , 197-218 . 257. , On some generalized Lebesgue functions and transformations, Colloq . Math . 5 4

(1987), no . 2 , 295-305 . 258. J . Haussermann , Generalized porosity characteristics of a residual set of continuous func-

tions, Ph . D . Thesis , U . California , Sant a Barbara , 1984 . 259. P . Humke an d M . Laczkovich, Typical continuous functions are virtually nonmonotone, Proc .

Amer. Math . Soc . 9 4 (1985) , 244-248 . 260. P . Humk e an d G . Petruska , The packing dimension of a typical function is two, Rea l Anal .

Ex. 1 4 (1988-89) , no . 2 , 345-358 . 261. B . Kaczmarski , On the set of points at which a function has an infinite one-sided derivative,

Real Anal . Ex . 1 6 (1990-91) , no . 2 , 421-424 . 262. L . I . Kapla n an d S . G . Slobodnik , Monotone transformations and differentiable properties

of functions, Mat . Zametk i 2 2 (1977) , no . 6 , 859-871 . (Russian ) 263. V . Kelar , On strict local extrema of differentiable functions, Rea l Anal . Ex . 6 (1980-81) ,

no. 2 , 242-244 . 264. B . Kirchheim, Typical approximately continuous functions are surprisingly thick, Rea l Anal .

Ex. 1 8 (1992-93) , no . 1 , 55-62 . 265. , Some further typical results on bounded Baire one functions, Act a Math . Hung . 6 2

(1993), no . 1-2 , 39-49 . 266. B . Kirchhei m an d T . Natkaniec , On universally bad Darboux functions, Rea l Anal . Ex . 1 6

(1990-91), no . 2 , 481-486 . 267. P . Komjath , A note on Darboux functions, Rea l Anal . Ex . 1 8 (1992-93) , no . 1 , 249-252 . 268. M . Laczkovich , Differentiable restrictions of continuous functions, Act a Math . Hung . 4 4

(1984), 355-360 . 269. M . Laczkovic h an d G . Petruska , Remarks on a problem of A. M. Bruckner, Act a Math .

Acad. Sci . Hung . 3 8 (1981) , 205-214 . 270. M . Laczkovic h an d D . Preiss , a-variation and transformation into C n functions, Rea l Anal .

Ex. 1 0 (1984-85) , no . 1 , 21-24 . 271. L . Larson , Symmetric real analysis: a survey, Rea l Anal . Ex . 9 (1983-84) , no . 1 , 154-178 . 272. V . Lebedev , Homeomorphisms of a closed interval and smoothness of a function, Mat . Za -

metki 4 0 (1986) , 364-373 ; Englis h transl . i n Math . Note s 4 0 (1986) . 273. R . Lewis , doctora l research , U . Californi a Sant a Barbara , 1988 , (unpublished) . 274. D . Mahara m an d A . Stone , Expressing measurable functions by one-to-one functions, Ad -

vances i n Math . 4 6 (1982) , 151-161 . 275. A . Maliszewski , Characteristic functions and products of bounded derivatives, Proc . Amer .

Math. Soc . (t o appear) . 276. J . Maly , Where the continuous functions without unilateral derivatives are typical, Trans .

Amer. Math . Soc . 28 3 (1984) , 169-175 .

SUPPLEMENTARY BIBLIOGRAPH Y 191

277. N . Manna an d S . Mukhopadhyay, On approximate derivates and their properties, Act a Math . Acad. Sci . Hung . 2 2 (1971) , 309-320 .

278. J . Mafik , Characteristic functions and products of derivatives, Rea l Anal . Ex . 1 6 (1990-91) , no. 1 , 245-256 .

279. , Derivatives and convexity, Rea l Anal . Ex . 1 7 (1991-92) , no . 2 , 745-747 . 280. J . Mafi k an d C . Weil , Products of powers of nonnegative derivatives, Trans . Amer . Math .

Soc. 27 6 (1983) , 361-373 . 281. , Sums of powers of derivatives, Proc . Amer . Math . Soc . 11 2 (1991) , no . 3 , 807-817 . 282. M . Matejdes , On the path derivative, Rea l Anal . Ex . 1 3 (1987-88) , no . 2 , 373-389 . 283. , Path differentiation in the Borel setting, Rea l Anal . Ex . 1 6 (1990-91) , no . 1 , 311-318 . 284. R . Mauldi n an d S . Williams , On the Hausdorff dimension of some graphs, Trans . Amer .

Math. Soc . 29 8 (1986) , 793-803 . 285. R . Menkyna , Classifying the set where a Baire one function is approximately continuous,

Real Anal . Ex . 1 4 (1988-89) , no . 2 , 413-419 . 286. A . Olevskii , Ulam-Zahorski problem on free interpretation by smooth functions, Trans . Amer .

Math. Soc . (t o appear) . 287. R . O'Malley , Decomposition of approximate derivatives, Proc . Amer . Math . Soc . 6 9 (1978) ,

234-247. 288. , Selective derivatives and the M2 or Denjoy-Clarkson properties, Act a Math . Acad .

Sci. Hung . 3 6 (1980) , 195-199 . 289. G . Petruska , Derivatives take every value in the set of approximate continuity points, Act a

Math. Hung . 4 2 (1983) , no . 3/4 , 355-360 . 290. D . Preiss , Maximoff's Theorem, Rea l Anal . Ex . 5 (1979-80) , no . 1 , 92-104 . 291. , Level sets of derivatives, Trans . Amer . Math . Soc . 27 2 (1982) , no . 1 , 161-184 . 292. , Algebra generated by derivatives, Rea l Anal . Ex . 8 (1982-83) , 205-214 . 293. D . Preis s an d B . Thomson, The approximate symmetric integral, Canad . J . Math . 4 1 (1989) ,

508-555. 294. D . Renfro , Some supertypical nowhere differentiability results for C[0, 1], Ph . D . Thesis ,

North Carolin a Stat e U. , 1993 . 295. T . Steele , doctora l research , U . Californi a Sant a Barbara , 1993 , (unpublished) . 296. K . Taylor , Darboux-like properties and monotonicity for generalized derivatives, Ph . D . The -

sis, U . California , Sant a Barbara , 1985 . 297. B . Thomson , Monotonicity theorems, Rea l Anal . Ex . 6 (1980-81) , no . 2 , 209-234 . 298. , Monotonicity theorems, Proc . Amer . Math . Soc . 8 3 (1981) , 847-852 . 299. , Derivation bases on the real line, I, Rea l Anal . Ex . 8 (1982-83) , no . 1 , 67-208 . 300. , Derivation bases on the real line, II, Rea l Anal . Ex . 8 (1982=83) , no . 2 , 278-442 . 301. , Symmetric Properties of Real Functions, Monograph s an d Textbook s i n Pur e an d

Applied Math. , Marce l Dekke r Inc . (t o appear) . 302. , Real Functions, Lectur e Note s i n Mathematics , vol . 1170 , Springer , 1985 . 303. R . Vallin , Shell porous, porous, totally porous, hyperporous, Rea l Anal . Ex . 1 8 (1992-93) ,

no. 2 , 294-320 . 304. C . Weil , The Peano derivative: What's known and what isn't, Rea l Anal . Ex . 9 (1983-84) ,

no. 2 , 354-365 . 305. H . Whitney , Analytic extensions of differentiate functions defined on cluster sets, Trans .

Amer. Math . Soc . 3 6 (1934) , 63-89 . 306. Wdjtowicz , The typical structure of the sets {x : f(x) = h(x)} for f continuous and h

Lipschitz, Trans . Amer . Math . Soc . 28 9 (1985) , 471-484 . 307. W . Young , Oscillating successions of continuous functions, Proc . Lond . Math . Soc . 6 (1908) ,

298-320. 308. Z . Zahorski , Sur Vensemble des points singuliere d'une fonction d'une variable reele admet-

tand les derivees de tous les ordres, Fund . Math . 3 4 (1947) , 183-245 . 309. L . Zajicek , On the symmetry of Dini derivates of arbitrary functions, Comm . Math . Univ .

Carol. 2 2 (1981) , 195-209 . 310. , Porosity and a-porosity, Rea l Anal . Ex . 1 3 (1987-1988) , no . 2 , 314-350 . 311. , Porosity, derived numbers and knot points of typical continuous functions, Czeck .

Math. J 3 9 (114 ) (1989) , 45-52 . 312. , Unpublished results of K. Pekdr and H. Zlonickd on preponderant derivatives and M4

sets, Rea l Anal . Ex . 1 5 (1989-90) , no . 1 , 413-418 .

Terminology Inde x

(Main reference s t o technica l terms )

approximate continuit y associated se t Banach's conditio n T 2 bilateral extrem e

derivate

congruent derivative s

Ch. 2 Ch. 7 Ch. 1 1

Ch. 4

Ch. 1 0 contraction o f a sequenc e

of interval s of set s of sets (regular )

convergent interva l function

Csaszar's derivativ e Darboux functio n

density density cove r

derivative o f Pompei u determining se t

5-differentiable

Dini derivative s External Intersectio n

Condition Garg's derivativ e

Ch. 2 Ch. 2

Ch. 2

Ch. 7 Ch. 1 0 Ch. 1

Ch. 2

Ch. 6 Ch. 2

Ch. 1 2

Ch. 1 5

Ch. 4

Ch. 1 5

Ch. 1 0

§5

§1 §2

§1 §4(d)

§5 §8

§8

§4 §4(c)

§5

§3 §6

§1

§1

§1 §4(h)

monotonic typ e nonangular

nonporous oscillation path derivativ e perfect roa d perfect syste m porosity preponderant continuit y preponderant derivativ e qualitative derivativ e reduction theore m

relative derivativ e restrictive set s

selective derivativ e cr-porous strong porosit y

symmetric derivativ e system o f path s stationary se t typical weakly differentiabl e

Ch. 1 3 § 1 Ch. 1 3 § 1

Ch. 1 5 § 1 Ch. 5 § 4 Ch. 1 5 § 1 Ch. 2 § 1 Ch. 7 § 2 Ch. 1 5 § 2 Ch. 1 0 § 4

Ch. 1 0 §4(a ) Ch. 1 0 §4(b ) Ch. 1 5 § 1 Ch. 1 0 §4(f )

Ch. 1 2 §2(b ) Ch. 1 0 §4(h )

Ch. 1 5 § 3

Ch. 1 5 § 2

Ch. 1 0 §4(e )

Ch. 1 5 § 1 Ch. 1 2 Ch. 1 3 Ch. 1 0 §4(h )

generalized absolut e

continuity Ch . 5 § 4 generalized bounde d

variation Ch . 5 § 4 Hamel basi s Ch . 1 § 1

maximal additiv e famil y Ch . 2 § 3 maximal multiplicativ e

family Ch . 2 § 3

193

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