- Title
- The differentiability of real functions on normed linear space using generalized subgradients
- Creator
- Borwein, J. M.; Fitzpatrick, S. P.; Giles, J. R.
- Relation
- Journal of Mathematical Analysis and Applications Vol. 128, Issue 2, p. 512-534
- Publisher Link
- http://dx.doi.org/10.1016/0022-247X(87)90203-4
- Publisher
- Academic Press
- Resource Type
- journal article
- Date
- 1987
- Description
- The modification of the Clarke generalized subdifferential due to Michel and Penot is a useful tool in determining differentiability properties for certain classes of real functions on a normed linear space. The Gâteaux differentiability of any real function can be deduced from the Gâteaux differentiability of the norm if the function has a directional derivative which attains a constant related to its generalized directional derivative. For any distance function on a space with uniformly Gâteaux differentiable norm, the Clarke and Michel-Penot generalized subdifferentials at points off the set reduce to the same object and this generates a continuity characterization for Gâteaux differentiability. However, on a Banach space with rotund dual, the Fréchet differentiability of a distance function implies that it is a convex function. A mean value theorem for the modified generalized subdifferential has implications for Gâteaux differentiability.
- Subject
- real functions; Gâteaux differentiability; generalized subgradients
- Identifier
- http://hdl.handle.net/1959.13/941053
- Identifier
- uon:13166
- Identifier
- ISSN:0022-247X
- Language
- eng
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