Fréchet directional differentiability and Fréchet differentiability

John R. Giles; Scott Sciffer

Commentationes Mathematicae Universitatis Carolinae (1996)

  • Volume: 37, Issue: 3, page 489-497
  • ISSN: 0010-2628

Abstract

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Zaj’ıček 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.

How to cite

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Giles, John R., and Sciffer, Scott. "Fréchet directional differentiability and Fréchet differentiability." Commentationes Mathematicae Universitatis Carolinae 37.3 (1996): 489-497. <http://eudml.org/doc/247871>.

@article{Giles1996,
abstract = {Zaj’ıček 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.},
author = {Giles, John R., Sciffer, Scott},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Gâteaux and Fréchet subdifferentiability; directional differentiability; strict and intermediate differentiability; Gâteaux and Fréchet subdifferentiability; directional differentiability; strict and intermediate differentiability},
language = {eng},
number = {3},
pages = {489-497},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Fréchet directional differentiability and Fréchet differentiability},
url = {http://eudml.org/doc/247871},
volume = {37},
year = {1996},
}

TY - JOUR
AU - Giles, John R.
AU - Sciffer, Scott
TI - Fréchet directional differentiability and Fréchet differentiability
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1996
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 37
IS - 3
SP - 489
EP - 497
AB - Zaj’ıček 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.
LA - eng
KW - Gâteaux and Fréchet subdifferentiability; directional differentiability; strict and intermediate differentiability; Gâteaux and Fréchet subdifferentiability; directional differentiability; strict and intermediate differentiability
UR - http://eudml.org/doc/247871
ER -

References

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  2. Contreras M.D., Paya R., On upper semi-continuity of duality mappings, Proc. Amer. Math. Soc. 121 (1994), 451-459. (1994) MR1215199
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  4. Giles J.R., Moors W.B., Generic continuity of restricted weak upper semi-continuous set-valued mappings, Set-valued Analysis, to appear. Zbl0852.54018MR1384248
  5. Giles J.R., Scott Sciffer, Continuity characterizations of differentiability of locally Lipschitz functions, Bull. Austral. Math. Soc. 41 (1990), 371-380. (1990) MR1071037
  6. Giles J.R., Scott Sciffer, Locally Lipschitz functions are generically pseudo-regular on separable Banach spaces, Bull. Austral. Math. Soc. 47 (1993), 203-210. (1993) MR1210135
  7. 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. (1958) MR0103243
  8. Phelps R.R., Convex Functions, Monotone Operators and Differentiability, Springer- Verlag, Lecture Notes in Math. 1364, 2nd ed., 1993. Zbl0921.46039MR1238715
  9. Zajíček L., Strict differentiability via differentiability, Act. U. Carol. 28 (1987), 157-159. (1987) MR0932752
  10. Zajíček L., Fréchet differentiability, strict differentiability and subdifferentiability, Czech. Math. J. 41 (1991), 471-489. (1991) MR1117801

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