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A note on the Darboux property of Fréchet derivatives. (English) Zbl 1130.26007

Let \(X\) be a real Banach space and \(B\) a nonempty subset of \(X\) such that \(\operatorname{int} B\) is connected and \(X\setminus \text{int} B\) is porous at every \(x\in B\cap\partial B\). Let \(f\) be a real-valued Fréchet differentiable function on \(B\). The authors prove that the graph of the derivative of \(f\) is a connected subset of \(X\times X^*\). In particular, the range of the derivative of \(f\) is a connected subset of \(X^*\). This generalizes a result of J. Malý [Real Anal. Exch. 22, No. 1, 167–173 (1996; Zbl 0879.26042)] where \(B\) is assumed to be a convex subset of \(X\) with nonempty interior.
Reviewer: Hans Weber (Udine)

MSC:

26B05 Continuity and differentiation questions
49J50 Fréchet and Gateaux differentiability in optimization

Citations:

Zbl 0879.26042
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Full Text: DOI Euclid