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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?

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Consider $f:\mathbb{R}^{2} \rightarrow \mathbb{R}$ with $f(x,y)=\mathrm{e}^{x}\cos y$ then $\nabla (f)$ is nothing but $\mathrm{e}^{\bar{z}} :\mathbb{C} \rightarrow \mathbb{C}$, with image neither convex nor simply connected. This gives a negative answer to the second and the third part of your question.

Regarding the first part I do not know the complete answer. But I can say only the following: for every $V\in \mathbb{R}^{n}$, $\nabla f[U]\cdot V$ is a connected subset of $\mathbb{R}$, because the partial derivatives satisfies Darboux theorem; hence they send open connected sets to connected subset of $\mathbb{R}$. Moreover, as a consequence of chain rule $\nabla f[U]\cdot V$ is a partial derivative. In fact there is no a hyper plane which separates $\nabla f[U]$.

So it is interesting to consider the following question:

Let $A$ be a subset of $R^{n}$, such that $A\cdot V$ is connected for all $V$, does this implies that $A$ is connected?

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    $\begingroup$ Consider the region $A\subset \mathbb{R}^n$, $n\geq 2$, defined as the union of the closed ball of radius $1/4$ centered at the origin with the annulus $3/4\leq \Vert x\Vert \leq 1$. It is disconnected and for any vector $V$ of length $1$ the projection $A\cdot V$ is the interval $[-1,1]$. $\endgroup$
    – Liviu Nicolaescu
    Dec 23, 2013 at 13:37
  • $\begingroup$ @liviuNicolaescu thanks for the example. However it can be shown that this set can not be equal to $\nabla f[U]$, when $U$ is open connected set. In fact no sphere can separates $\nabla f[U]$. Without lose of generality assume the sphere which separates the image, is the unit sphere around 0. Let $a,b \in U$ and $\parallel \nabla f(a)\parallel <1$ and $\parallel \nabla f(b)\parallel >1$. Choose a unit speed curve $\gamma: [0,1] \rightarrow U$ which connect a to b and its velocity at end points is parallel to $\nabla f(a)$, $\nabla f(b)$ now apply Darboux to $f\circ \gamma$ $\endgroup$
    – Ali Taghavi
    Dec 23, 2013 at 18:43
  • $\begingroup$ So it is natural to ask:"Let $A$ be a subset of $\mathbb{R}^{n}$ which can be separated by no hyperplane or sphere, does it implies that $A$ is connected"? $\endgroup$
    – Ali Taghavi
    Dec 23, 2013 at 18:48
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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

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.

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  • $\begingroup$ I didn't get why that function is being a counterexample for generalization of Darboux's theorem ! can you explainit? $\endgroup$
    – Red shoes
    May 28, 2017 at 5:13
  • $\begingroup$ @Ashkan Are you talking about the function in the linked paper "Solution to the gradient..."? $\endgroup$
    – Ali Taghavi
    Jul 10, 2017 at 14:27
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    $\begingroup$ @Ali Taghavi. YES , function with above property $\endgroup$
    – Red shoes
    Jul 10, 2017 at 14:36
  • $\begingroup$ @Ashkan Very interesting question! If I understand your question correctly, you say why does this f is a counter example to the first part of the OP's question. yes? This $f$ is a counter example if that zero measure set would be a countable set?But is it really the case? $\endgroup$
    – Ali Taghavi
    Jul 10, 2017 at 17:02

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