## Real Analysis Exchange

### A Note on the Darboux Property of Fréchet Derivatives

#### Abstract

et $A$ be a subset of a Banach space $X$ and $f$ a Fréchet differentiable function on $A$ (with respect to $A$). We give a simple proof of the connectedness of the graph of $f'$ in $X\times X^*$ under relatively weak conditions on $A$. In particular, we simplify a proof by J. Malý of the connectedness of the range of $f'$ for some convex sets $A$. At the same time, we extend an older result of C. E. Weil on the connectedness of the range of $f'$ for some non-convex sets $A\subset\mathbb R^n$.

#### Article information

Source
Real Anal. Exchange, Volume 32, Number 2 (2006), 489-494.

Dates
First available in Project Euclid: 3 January 2008

https://projecteuclid.org/euclid.rae/1199377485

Mathematical Reviews number (MathSciNet)
MR2369857

Zentralblatt MATH identifier
1130.26007

Subjects
Primary: 26B06

#### Citation

Holický, P.; Weil, C. E.; Zajíček, L. A Note on the Darboux Property of Fréchet Derivatives. Real Anal. Exchange 32 (2006), no. 2, 489--494. https://projecteuclid.org/euclid.rae/1199377485