Characterization of the upper subderivative and its consequences in nonsmooth analysis (Q1192398)

Let \(X\) be a Banach space, and let \(f\) be an extended-real-valued lower semicontinuous function on \(X\). It is proved that \[ \limsup_{{u\to x\atop f(u)\to f(x)}} f^ \# (u,y)= f^ \uparrow(x,y), \] where \(f^ \uparrow\) is the upper subderivative of \(f\) [see \textit{R. T. Rockafellar}, Proc. Lond. Math. Soc., III. Ser. 39, 331-355 (1979; Zbl 0413.49015)], and \(f^ \#\) is the contingent directional derivative of \(f\) [see \textit{J. P. Aubin} and \textit{I. Ekeland}, `Applied nonlinear analysis' (1984; Zbl 0641.47066)]. This result is then applied to extend some known theorems of nonsmooth analysis from a finite-dimensional space to an infinite-dimensional Banach space.