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Characterization of the upper subderivative and its consequences in nonsmooth analysis. (English) Zbl 0795.49017
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 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 J. P. Aubin and 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.
##### MSC:
 49J52 Nonsmooth analysis 46G05 Derivatives of functions in infinite-dimensional spaces