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non-zero gradient

Proceedings of the Edinburgh Mathematical Society (1)

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A note on a Residual Subset of Lipschitz Functions on Metric Spaces
Fabio Cavalletti
Journal:

Proceedings of the Edinburgh Mathematical Society / Volume 58 / Issue 3 / October 2015

Published online by Cambridge University Press:

30 December 2014, pp. 631-636
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Let (X, d) be a quasi-convex, complete and separable metric space with reference probability measure m. We prove that the set of real-valued Lipschitz functions with non-zero pointwise Lipschitz constant m-almost everywhere is residual, and hence dense, in the Banach space of Lipschitz and bounded functions. The result is the metric analogous to a result proved for real-valued Lipschitz maps defined on ℝ2 by Alberti et al.