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The Darboux property for gradients. (English) Zbl 0879.26042
Let $$X$$ be a Banach space, $$X^{*}$$ its dual and $$f$$ a Fréchet differentiable function on $$D\subset X$$. The author proves that the derivative $$f'$$ of $$f$$ has the Darboux property in the sense that $$f'(K)$$ is a connected subset of $$X^{*}$$ whenever $$K\subset D$$ is closed, convex, and $$\text{Int } K\neq \emptyset$$. This especially includes gradients of functions of several variables; for the latter case see C. J. Neugebauer [Trans. Am. Math. Soc. 107, 30-37, (1963; Zbl 0112.04003)] and C. E. Weil [Pac. J. Math. 44, 757-765 (1973; Zbl 0258.26006)]. There is an example in the paper, showing that the assumption about the interior of $$K$$ is essential.
Reviewer: M.Krbec (Praha)

##### MSC:
 26B05 Continuity and differentiation questions 46G05 Derivatives of functions in infinite-dimensional spaces