| Title:
|
Fréchet directional differentiability and Fréchet differentiability (English) |
| Author:
|
Giles, J. R. |
| Author:
|
Sciffer, Scott |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
37 |
| Issue:
|
3 |
| Year:
|
1996 |
| Pages:
|
489-497 |
| . |
| Category:
|
math |
| . |
| Summary:
|
Zaj'\i ček has recently shown that for a lower semi-continuous real-valued function on an Asplund space, the set of points where the function is Fréchet subdifferentiable but not Fréchet differentiable is first category. We introduce another variant of Fréchet differentiability, called Fréchet directional differentiability, and show that for any real-valued function on a normed linear space, the set of points where the function is Fréchet directionally differentiable but not Fréchet differentiable is first category. (English) |
| Keyword:
|
G\^ateaux and Fréchet subdifferentiability |
| Keyword:
|
directional differentiability |
| Keyword:
|
strict and intermediate differentiability |
| MSC:
|
46G05 |
| MSC:
|
58C20 |
| idZBL:
|
Zbl 0881.58011 |
| idMR:
|
MR1426913 |
| . |
| Date available:
|
2009-01-08T18:25:34Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118855 |
| . |
| Reference:
|
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| Reference:
|
[CP] Contreras M.D., Paya R.: On upper semi-continuity of duality mappings.Proc. Amer. Math. Soc. 121 (1994), 451-459. MR 1215199 |
| Reference:
|
[FP] Fabian M., Preiss D.: On intermediate differentiability of Lipschitz functions on certain spaces.Proc. Amer. Math. Soc. 113 (1991), 733-740. MR 1074753 |
| Reference:
|
[GM] Giles J.R., Moors W.B.: Generic continuity of restricted weak upper semi-continuous set-valued mappings.Set-valued Analysis, to appear. Zbl 0852.54018, MR 1384248 |
| Reference:
|
[GS1] Giles J.R., Scott Sciffer: Continuity characterizations of differentiability of locally Lipschitz functions.Bull. Austral. Math. Soc. 41 (1990), 371-380. MR 1071037 |
| Reference:
|
[GS2] Giles J.R., Scott Sciffer: Locally Lipschitz functions are generically pseudo-regular on separable Banach spaces.Bull. Austral. Math. Soc. 47 (1993), 203-210. MR 1210135 |
| Reference:
|
[M] Marcus S.: Sur les fonctions dérivées, intégrables, au sens de Riemann et sur les dérivées partialles mixtes.Proc. Amer. Math. Soc. 9 (1958), 973-978. MR 0103243 |
| Reference:
|
[Ph] Phelps R.R.: Convex Functions, Monotone Operators and Differentiability.Springer- Verlag, Lecture Notes in Math. 1364, 2nd ed., 1993. Zbl 0921.46039, MR 1238715 |
| Reference:
|
[Z1] Zajíček L.: Strict differentiability via differentiability.Act. U. Carol. 28 (1987), 157-159. MR 0932752 |
| Reference:
|
[Z2] Zajíček L.: Fréchet differentiability, strict differentiability and subdifferentiability.Czech. Math. J. 41 (1991), 471-489. MR 1117801 |
| . |
