**Darboux's Theorem.** *If $f:[a,b]\to\mathbb R$ is differentiable and $f'(a)<\xi<f'(b)$, then there exists a $c\in (a,b)$, such that $\,f'(c)=\xi$.*


Does any of the following generalizations


1. Let $U\subset\mathbb R^n$ connected and $f: U\to \mathbb R$ differentiable. Then $\nabla f[U]$ is connected,
2. Let $U\subset\mathbb R^n$ convex and $f: U\to \mathbb R$ differentiable. Then $\nabla f[U]$ is convex,
3. $H\_k\big(\nabla f[U],\mathbb{Z}\big) \hookrightarrow H\_k(U,\mathbb{Z})$, for all $k$,


hold?



Generalization of Darboux's Theorem
It turns out than none of the three potential generalisations holds.


Counterexamples for the last two questions are presented in the answer of Ali Taghavi, and in particular by function $f(x,y)=(\mathrm{e}^x\cos y,\mathrm{e}^x\sin y)$, as $f[\mathbb R^2]=\mathbb R^2\smallsetminus\{(0,0)\}$.


For the first question, a counterexample appears in:


[Solution to the gradient problem of C.E. Weil, by Zoltán Buczolich](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.rmi/1136999135)


where the author gives a complete answer to the famous gradient problem of C. E. Weil. On an open set $G\subset \mathbb{R}^{2}$ he constructs a differentiable function $f:G\to\mathbb{R}$, for which there exists an open set $\Omega\_{1}\subset\mathbb{R}^{2}$ such that $\nabla f({p})\in \Omega\_{1}$ for a ${p}\in G$ but $\nabla f({q})\not\in\Omega\_{1}$ for almost every ${q}\in G$. This also shows that the Denjoy-Clarkson property does not hold in higher dimensions.


