Fréchet differentiability of regular locally Lipschitzian functions (Q1176960)

Locally Lipschitzian regular (i.e. the one sided directional derivative exists and equals the generalized directional derivative) real-valued functions defined on open subsets of separable Banach spaces are considered. For any such function it is shown that Clarke's generalized gradient is a minimal, convex and compact valued upper semicontinuous multi-function. Using a theorem of \textit{J. P. R. Christensen} and \textit{P. S. Kenderov} [Math. Scand. 54, 70-78 (1984; Zbl 0557.46016)] it is then shown that for separable Asplund spaces such a function is Fréchet differentiable on a dense \(G_ \delta\) subset of its domain. This should be compared with a result of \textit{D. Preiss} [J. Funct. Anal. 91, No. 2, 312-345 (1990; Zbl 0711.46036)], namely that any locally Lipschitzian real-valued function on a not necessarily separable Banach space is Fréchet differentiable on a dense set. However they give an example showing that the regularity assumption cannot be dropped.