Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Differentiability via directional derivatives


Authors: Ka Sing Lau and Clifford E. Weil
Journal: Proc. Amer. Math. Soc. 70 (1978), 11-17
MSC: Primary 26A24; Secondary 58C20
DOI: https://doi.org/10.1090/S0002-9939-1978-0486354-7
MathSciNet review: 0486354
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let F be a continuous function from an open subset D of a separable Banach space X into a Banach space Y. We show that if there is a dense $ {G_\delta }$ subset A of D and a $ {G_\delta }$ subset H of X whose closure has nonempty interior, such that for each $ a \in A$ and each $ x \in H$ the directional derivative $ {D_x}F(a)$ of F at a in the direction x exists, then F is Gâteaux differentiable on a dense $ {G_\delta }$ subset of D. If X is replaced by $ {R^n}$, then we need only assume that the n first order partial derivatives exist at each $ a \in A$ to conclude that F is Frechet differentiable on a dense, $ {G_\delta }$ subset of D.


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

  • [1] J. Christensen, Topology and Borel structure, North-Holland, Amsterdam; American Elsevier, New York, 1973. MR 0348724 (50:1221)
  • [2] H. Federer, Geometric measure theory, Springer-Verlag, New York, 1969. MR 0257325 (41:1976)
  • [3] M. Fort, Jr., Category theorems, Fund. Math. 42 (1955), 276-288. MR 0077911 (17:1115b)
  • [4] P. Mankiewicz, On the differentiability of Lipschitz mappings in Fréchet spaces, Studia Math. 45 (1973), 15-29. MR 0331055 (48:9390)
  • [5] -, On topological, Lipschitz and uniform classification of LF-spaces, Studia Math. 52 (1974), 107-142. MR 0402448 (53:6268)
  • [6] H. Rademacher, Über partielle und totale Differenzierbarkeit, Math. Ann. 79 (1919), 340-359. MR 1511935
  • [7] W. Stepanoff, Sur le conditions de l'existence de la differentielle totale, Rec. Math. Soc. Math. Moscow 32 (1925), 511-526.
  • [8] Z. Zahorski, Sur l'ensemble des points de non dérivabilité d'une fonction continue, Bull. Soc. Math. France 74 (1946), 147-178. MR 0022592 (9:231a)
  • [9] S. Yamamuro, Differential calculus in topological linear spaces, Lecture Notes in Math., no. 374, Springer-Verlag, Berlin and New York, 1974. MR 0488118 (58:7686)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 26A24, 58C20

Retrieve articles in all journals with MSC: 26A24, 58C20


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1978-0486354-7
Keywords: Baire space, dense $ {G_\delta }$, directional, Gâteaux, Frechet derivatives
Article copyright: © Copyright 1978 American Mathematical Society

American Mathematical Society