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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)

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