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Distinct differentiable functions may share the same Clarke subdifferential at all points
Author(s):
J.
M.
Borwein;
Xianfu
Wang
Journal:
Proc. Amer. Math. Soc.
125
(1997),
807-813.
MSC (1991):
Primary 49J52;
Secondary 26A27, 26A16
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Abstract:
We construct, using Zahorski's Theorem, two everywhere differentiable real-valued Lipschitz functions differing by more than a constant but sharing the same Clarke subdifferential and the same approximate subdifferential.
References:
- 1.
- A. M. BRUCKNER, Differentiation of Real Functions, Lecture Notes in Mathematics edited by A. Dold and B. Eckmann, Springer-Verlag, 1978. MR 80h:26002
- 2.
- A. M. BRUCKNER AND G. PETRUSKA, Some Typical Results on Bounded Baire-1 Functions, Acta Math. Hung. 43 (3-4) (1984), 325-333. MR 85h:26004
- 3.
- F. H. CLARKE, Optimization and Nonsmooth Analysis, Wiley Interscience, New York, 1983. MR 85m:49002
- 4.
- A. D. IOFFE, Approximate Subdifferentials and Applications 3: The Metric Theory, Mathematika Vol. 36, No. 71 (1989), 1-38. MR 90g:49012
- 5.
- -, Approximate Subdifferentials and Applications I: The Finite Dimensional Theory, Transactions of The American Mathematical Society 281 (1984), 390-416. MR 84m:49029
- 6.
- P. MICHEL AND J. P. PENOT, Calcul Sous-différentiel Pour les Fonctions Lipschitzienne et non Lipschitzienne, C. R. Acad. Sci. Paris 298 (1984), 269-272. MR 85i:49027
- 7.
- H. L. ROYDEN, Real Analysis, Macmillan Publishing Company, New York, 1988. MR 90g:00004
- 8.
- K. R. STROMBERG, An Introduction to Classical Real Analysis, Wadsworth International Mathematics Series, 1981. MR 82c:26002
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Additional Information:
J.
M.
Borwein
Affiliation:
Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
Email:
jborwein@cecm.sfu.ca
Xianfu
Wang
Affiliation:
Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
Email:
xwang@cecm.sfu.ca
DOI:
10.1090/S0002-9939-97-03654-X
PII:
S 0002-9939(97)03654-X
Keywords:
Lipschitz function,
differentiability,
integrability,
generalized derivative,
Clarke subdifferential,
approximate continuity,
metric density
Received by editor(s):
July 13, 1995
Received by editor(s) in revised form:
September 8, 1995
Additional Notes:
The first author's research supported by NSERC and the Shrum Endowment at Simon Fraser University.
Communicated by:
J. Marshall Ash
Copyright of article:
Copyright
1997,
American Mathematical Society
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