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?

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?

  • 1
    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 $[-1,1]$. – Liviu Nicolaescu Dec 23 '13 at 13:37
  • @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 $f\circ \gamma$ – Ali Taghavi Dec 23 '13 at 18:43
  • 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"? – Ali Taghavi Dec 23 '13 at 18:48
up vote 11 down vote accepted

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.

  • I didn't get why that function is being a counterexample for generalization of Darboux's theorem ! can you explainit? – Red shoes May 28 '17 at 5:13
  • @Ashkan Are you talking about the function in the linked paper "Solution to the gradient..."? – Ali Taghavi Jul 10 '17 at 14:27
  • 1
    @Ali Taghavi. YES , function with above property – Red shoes Jul 10 '17 at 14:36
  • @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? – Ali Taghavi Jul 10 '17 at 17:02

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