Abstract
We prove that the Rieffel sharpness condition for a Banach space E is necessary and sufficient for an arbitrary Lipschitz function f: [a, b]→E to be differentiable almost everywhere on a segment [a, b]. We establish that, in the case where the sharpness condition is not satisfied, the major part (in the category sense) of Lipschitz functions has no derivatives at any point of the segment [a, b].
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Additional information
Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 4, pp. 500–509, April, 1997.
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Bondar’, A.V. On differential properties of mappings into a Banach space. Ukr Math J 49, 550–560 (1997). https://doi.org/10.1007/BF02487317
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DOI: https://doi.org/10.1007/BF02487317