Timeline for Generalization of Darboux's Theorem

Current License: CC BY-SA 3.0

11 events

when toggle format	what		by	license	comment
Dec 25, 2013 at 22:46	vote	accept	smyrlis
Dec 25, 2013 at 22:46
Dec 24, 2013 at 8:17	history	edited	Ali Taghavi	CC BY-SA 3.0	added 5 characters in body
Dec 23, 2013 at 18:48	comment	added	Ali Taghavi		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"?
Dec 23, 2013 at 18:43	comment	added	Ali Taghavi		@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 $f\circ \gamma$
Dec 23, 2013 at 13:37	comment	added	Liviu Nicolaescu		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 $[-1,1]$.
Dec 22, 2013 at 18:30	history	edited	Ali Taghavi	CC BY-SA 3.0	deleted 3 characters in body
S Dec 22, 2013 at 16:49	history	suggested	smyrlis	CC BY-SA 3.0	Minor grammatical, spelling and LaTeX corrections.
Dec 22, 2013 at 16:46	review	Suggested edits
S Dec 22, 2013 at 16:49
Dec 22, 2013 at 16:13	history	edited	Ali Taghavi	CC BY-SA 3.0	added 54 characters in body
Dec 22, 2013 at 15:55	history	edited	Ali Taghavi	CC BY-SA 3.0	added 67 characters in body
Dec 22, 2013 at 15:49	history	answered	Ali Taghavi	CC BY-SA 3.0