We show that every Lipschitz map defined on an open subset of the Banach space C(K), where K is a scattered compactum, with values in a Banach space with the Radon-Nikodym property, has a point of Fréchet differentiability. This is a strengthening of the result of Lindenstrauss and Preiss who proved that for countable compacta. As a consequence of the above and a result of Arvanitakis we prove that Lipschitz functions on certain function spaces are Gâteaux differentiable.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba58-3-8,
author = {Rafa\l\ G\'orak},
title = {A Note on Differentiability of Lipschitz Maps},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
volume = {58},
year = {2010},
pages = {259-268},
zbl = {1214.46025},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba58-3-8}
}
Rafał Górak. A Note on Differentiability of Lipschitz Maps. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 58 (2010) pp. 259-268. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba58-3-8/