Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Path derivatives: a unified view of certain generalized derivatives


Authors: A. M. Bruckner, R. J. O’Malley and B. S. Thomson
Journal: Trans. Amer. Math. Soc. 283 (1984), 97-125
MSC: Primary 26A24
DOI: https://doi.org/10.1090/S0002-9947-1984-0735410-1
MathSciNet review: 735410
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A collection $ E = \{ {E_x}:x \in R\} $ is a system of paths if each set $ {E_x}$ has $ x$ as a point of accumulation. For such a system $ E$ the derivative $ F_E'(x)$ of a function $ F$ at a point $ x$ is just the usual derivative at $ x$ relative to the set $ {E_x}$. The goal of this paper is the investigation of properties that $ F$ and its derivative $ F_E'$ must have under certain natural assumptions about the collection $ E$. In particular, it is shown that most of the familiar properties of approximate derivatives and approximately differentiable functions follow in this setting from three conditions on the collection $ E$ relating to the "thickness" of the sets $ {E_x}$ and the way in which the sets intersect.


References [Enhancements On Off] (What's this?)

  • [1] S. Agronsky, R. Biskner, A. Bruckner and J. Mařik, Representations of functions by derivatives, Trans. Amer. Math. Soc. 263 (1981), 493-500. MR 594421 (82e:26006)
  • [2] C. L. Belna, M. J. Evans and P. D. Humke, Symmetric and ordinary differentiation, Proc. Amer. Math. Soc. 72 (1978), 261-267. MR 507319 (80d:26006)
  • [3] A. M. Bruckner, Some observations about Denjoy's preponderant derivative, Bull. Math. Soc. Sci. Math. R. S. Roumaine (N.S.) 21 (69) (1977), 2-10. MR 0470158 (57:9922)
  • [4] -, Differentiation of real functions, Lecture Notes in Math., vol. 659, Springer, Berlin and New York, 1978. MR 507448 (80h:26002)
  • [5] -, On the differentiation of integrals in euclidean spaces, Fund. Math. 66 (1969), 129-135. MR 0262430 (41:7037)
  • [6] A. Bruckner and C. Goffman, The boundary behaviour of real functions in the upper half-plane, Rev. Roumaine Math. Pures Appl. 11 (1966), 507-518. MR 0206173 (34:5995)
  • [7] -, Approximate differentiation, Real Anal. Exchange 6 (1980/81), 9-65. MR 606541 (82d:26004)
  • [8] H. Croft, A note on a Darboux continuous function, J. London Math. Soc. 38 (1963), 9-10. MR 0147588 (26:5103)
  • [9] A. Denjoy, Sur les fonctions dérivées sommables, Bull. Soc. Math. France 43 (1915), 161-248. MR 1504743
  • [10] -, Sur une propriété des fonctions dérivées, Enseignement Math. 18 (1916), 320-328.
  • [11] E. P. Dolženko, Boundary properties of real functions, Math. USSR-Izv. 1 (1967), 1-12.
  • [12] H. W. Ellis, Darboux properties and applications to non-absolutely convergent integrals, Canad. J. Math. 3 (1951), 471-484. MR 0043872 (13:332d)
  • [13] M. J. Evans and P. D. Humke, The equality of unilateral derivates, Proc. Amer. Math. Soc. 79 (1980), 609-613. MR 572313 (81h:26002)
  • [14] A. Gleyzal, Interval functions, Duke Math. J. 8 (1941), 223-230. MR 0005891 (3:226f)
  • [15] C. Goffman and C. Neugebauer, On approximate derivatives, Proc. Amer. Math. Soc. 11 (1960), 962-966. MR 0118792 (22:9562)
  • [16] -, Linear continuous functions, Proc. Amer. Math. Soc. 12 (1961), 997-998. MR 0136686 (25:151)
  • [17] V. Jarník, Über die Differenzierbarkeit stetizer Funktionen, Fund. Math. 21 (1933), 48-58.
  • [18] A. Khintchine, Recherches sur la structure des fonctions mesurables, Fund. Math. 9 (1927), 212-279.
  • [19] M. Laczkovich, On the Baire class of selective derivatives, Acta. Math. Sci. Hungar. 29 (1977), 99-105. MR 0437691 (55:10615)
  • [20] S. Marcus, La dérivée approximative qualitative, Com. Acad. R. P. Romaine 3 (1953), 361 -364. MR 0074481 (17:593d)
  • [21] M. Mastalerz-Wawrzńczak, On a certain condition of the monotonicity of functions, Fund. Math. 97 (1977), 187-198. MR 0453943 (56:12196)
  • [22] L. Mišík, Über approximative derivierte Zahlen monotoner Funktionen, Czechoslovak Math. J. 26 (101) (1976), 579-583. MR 0432834 (55:5814)
  • [23] H. Oliver, The exact Peano derivative, Trans. Amer. Math. Soc. 76 (1954), 444-456. MR 0062207 (15:944d)
  • [24] R. J. O'Malley, Baire$ ^{*}$ $ 1$, Darboux functions, Proc. Amer. Math. Soc. 60 (1976), 187-192.
  • [25] -, Selective derivates, Acta. Math. Acad. Sci. Hungar. 29 (1977), 77-97. MR 0437690 (55:10614)
  • [26] -, Decomposition of approximate derivatives, Proc. Amer. Math. Soc. 69 (1978), 243-247. MR 0466446 (57:6325)
  • [27] -, Selective derivatives and the $ {M_2}$ or Denjoy-Clarkson properties, Acta Math. Acad. Sci. Hungar. 36 (1980), 195-199. MR 605190 (82b:26009)
  • [28] -, Selective differentiation: Redefining selections (submitted).
  • [29] -, Bi-selective derivatives are of honorary Baire class $ 2$ (submitted).
  • [30] R. J. O'Malley and C. E. Weil, The oscillatory behaviour of certain derivatives, Trans. Amer. Math. Soc. 234 (1977), 467-481. MR 0453940 (56:12193)
  • [31] -, Composite differentiation (in preparation)
  • [32] G. Petruska and M. Laczkovitch, Remarks on a problem of A. M. Bruckner, Acta. Math. Acad. Sci. Hungar. 38 (1981), 205-214. MR 634581 (83b:26005)
  • [33] S. Saks, Theory of the integral Monografie Mat. 7, PWN, Warszawa, 1937.
  • [34] J. Scholz, Essential derivations of functions in $ C[a,b]$ (to appear).
  • [35] G. Sindalovski, Derivatives of continuous functions, Izv. Akad. Nauk SSSR 32 (1968). (Russian)
  • [36] T. Światkowski, On a certain generalization of the notation of the derivative (in Polish), Zeszyty Nauk. Politech. Łódz. Mat. No. 149 (1972), 89-103.
  • [37] G. Tolstoff, Sur la dérivée approximative exacte, Rec. Math. (Mat. Sbornik) N.S. (1938), 499-504.
  • [38] B. S. Thomson, Monotonicity theorems, Real Anal. Exchange 6 (1980/81), 209-234. MR 623052 (83b:26015)
  • [39] -, Monotonicity theorems, Proc. Amer. Math. Soc. 83 (1981), 547-552. MR 627688 (83b:26014)
  • [40] S. Verblunsky, On the Peano derivatives, Proc. London Math. Soc. (3) 22 (1971), 313-324. MR 0285678 (44:2896)
  • [41] C. Weil, On properties of derivatives, Trans. Amer. Math. Soc. 114 (1965), 363-376. MR 0176007 (31:283)
  • [42] -, On approximate and Peano derivatives, Proc. Amer. Math. Soc. 20 (1969), 487-490. MR 0233944 (38:2265)
  • [43] -, A property for certain derivatives, Indiana Univ. Math. J. 23 (1973/74), 527-536. MR 0335703 (49:483)
  • [44] G. C. Young, On the derivates of a function, Proc. London Math. Soc. (2) 15 (1916), 360-384.
  • [45] W. H. Young, Oscillating successions of continuous functions, Proc. London Math. Soc. 6 (1908), 805-807.
  • [46] Z. Zahorski, Sur la première dérivée, Trans. Amer. Math. Soc. 69 (1950), 1-54. MR 0037338 (12:247c)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 26A24

Retrieve articles in all journals with MSC: 26A24


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1984-0735410-1
Keywords: Derivatives, approximate derivatives, Peano derivatives, functions of Baire class $ 1$, Darboux functions
Article copyright: © Copyright 1984 American Mathematical Society

American Mathematical Society