Csornyei, M and Holicky, P (2008) Darboux property of gateaux derivatives of functions on R-n. ACTA MATH HUNG , 120 (3) 209 - 234. 10.1007/s10474-008-6181-3.
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Abstract
D. Preiss proved that the graph of the derivative of a continuous Gateaux-differentiable function f : R-2 -> R is always connected. We show that this is no longer true in higher dimensions: we construct a continuous; Gateaux-differentiable function f : R-3 -> R for which the range of its gradient mapping {del f (x) : x epsilon R-3} is disconnected. We also give an example of an approximately differentiable continuous function on R-2 such that the range of its gradient mapping is disconnected.
| Type: | Article |
|---|---|
| Title: | Darboux property of gateaux derivatives of functions on R-n |
| DOI: | 10.1007/s10474-008-6181-3 |
| Keywords: | gateaux derivative, approximate derivative, Darboux properties, connected graph, BANACH-SPACES, RANGE |
| UCL classification: | UCL > School of BEAMS > Faculty of Maths and Physical Sciences > Mathematics |
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