Does a compact embedded manifold achieve all unit normals?
Prove that Gauss map on M is surjective
Is the function differentiable at $0$? Let $$f(x) = \begin{cases}\begin{align*}&amp;\cos{\dfrac{1}{x}}, &amp;x \neq0 \\ &amp;0, &amp;x=0. \end{align*}\end{cases}$$ Is the function $F(x) = \displaystyle \int_{0}^x f dx$ differentiable at $0$?  We can see that the function $f(x)$ is continuous everywhere except $x=0$. In order to show differentiability we will need to show the derivatives from the left and right are equal. So we need to show that $$\lim_{h \to 0^+} \dfrac{F(x+h)-F(x)}{h} = \lim_{h \to 0^-} \dfrac{F(x+h)-F(x)}{h}.$$ The derivative from the right is $$\lim_{h \to 0^+} \dfrac{F(x+h)-F(x)}{h} = \lim_{h \to 0^+} \dfrac{\displaystyle \int_{0}^x\cos{\dfrac{1}{x+h}}-\int_{0}^x\cos{\dfrac{1}{x}}}{h}$$ and the derivative from the left is $$\lim_{h \to 0^-} \dfrac{F(x+h)-F(x)}{h} = \lim_{h \to 0^-} \dfrac{\displaystyle \int_{0}^x\cos{\dfrac{1}{x+h}}-\int_{0}^x\cos{\dfrac{1}{x}}}{h}.$$ I am not sure how to proceed. Of Half-elven and mortal life WordPress Installation Failed I am trying to install Wordpress on Lubuntu. I followed the instructions . Including PHP Apache and MySQL setups. Apache and MySQL appear to be running fine.  However, when I go to http://localhost/blog/, I get the following error:     Neither /etc/wordpress/config-localhost.php nor /etc/wordpress/config-localhost.php could be found.   Ensure one of them exists, is readable by the webserver and contains the right password/username. An inequality for a sum of sine functions I want to show for $x\in\mathbb{R}$, the following inequality holds. $$ \left|\sin x\right|+\left|\sin (x+1)\right|+\left|\sin (x+2)\right|&gt;8/5. $$ This trigonometric inequality has been verified by Mathematica using the Plot commend. However, using the sum formula for the sine function seems doesn't work, and I cannot give a rigorous proof of it. Any suggestion, idea, or comment is welcome, thanks! Group theory proof of existence of a solution to $x^2\equiv -1\pmod p$ iff $p\equiv 1 \pmod 4$ I've read through the elementary proof of why there exists a solution $x$ to $x^2\equiv -1\pmod p$ iff $p\equiv 1 \pmod 4$ for $p$ an odd prime. Is there a group theory generalization for this fact as well? The thing with these tags is, you can always (or almost always) use a specific product/technology/language tag which would render the company-name tag completely redundant (see Appendix C for sloppy non-proof). For example, what relevant information would  tag add to  already tagged with ? What would  tag add to  already tagged with ? Another way of looking at this: how much useful information do you lose if you don't tag a question with a company name? Note: I'm not talking about brand/company/organizations like , , ,  because those names also refer to websites/web services with which you can interact via API or from which you can extract data. Appendix C: The &quot;about&quot; test The relationship between a tag and a question can be expressed as such  This question is about [tag]  In case of ,  This question is about [microsoft]  Fair enough. But in reality people are using these company-name tags like this  This question is about something developed by or related to [microsoft]  But if that's the case, you might as well just go with  This question is about [something developed by or related to microsoft]  where  can be , , ... you get the idea. 101 positive integers placed on a circle A Pigeonhole Principle problem:     101 positive integers are placed on a circle whose sum is 300. Prove   that it is possible to choose from these numbers some consecutive   numbers whose sum is equal to 200.   (I don't know if the word 'consecutive' is appropriate in this case ,I mean that these numbers follow each other on that circle) How to clear bash history completely? Windows Explorer sort method I'm looking for an algorithm that sorts strings similar to the way files (and folders) are sorted in Windows Explorer. It seems that numeric values in strings are taken into account when sorted which results in something like  name 1, name 2, name 10   instead of  name 1, name 10, name 2   which you get with a regular string comparison.  I was about to start writing this myself but wanted to check if anyone had done this before and was willing to share some code or insights. The way I would approach this would be to add leading zeros to the numeric values in the name before comparing them. This would result in something like  name 00001, name 00010, name 00002   which when sorted with a regular string sort would give me the correct result.  Any ideas? How do you challenge the closing of one of your questions? As a newcomer to StackOverflow, I recently asked my first . I wanted to start with a fairly safe general question, to get the feel for how the site worked. My question was intended to be programming-related, but my original wording was not clear enough, so another user decided to close it.  I've re-phrased my question, and had received one useful answer before the question was closed, but I'm not sure what the procedure is after your question is closed to allow people to post answers again...  I've checked the FAQ and About pages, searched the SO questions, but have had no luck. Could someone point me in the right direction as to how I get my question re-opened, or if it is possible to do at all? Windows Activation FAQ: How do language, version, 64-bit or 32-bit, and source affect ability to install and transfer Windows licenses? We get a lot of questions on Super User about Windows activation and licensing. This question is an attempt to answer the most common questions in one place with a .   When reinstalling Windows, the following questions often come up:   Can I transfer the copy of Windows from my old computer to a new one? Does the license key on the sticker on my computer work on any computer? If I have a 32-bit version, can I install a 64-bit version? I have an OEM version of Windows, but only a retail disk. Can I still use it? I want to install Windows with a different language. Do I need a new license in that language? I changed some hardware. Will I be able to reactivate my computer? Can I back up my activation before doing a reinstall? How many computers can I install my license on? Do I have to deactivate my old installation before transferring a license? If so, how? How does licensing work in Windows ? What are the restrictions on transferring, reinstalling, and changing versions of Windows? Most instructions for equipment and software make a habit of shortening sentences by dropping words of certain parts of speech - usually definite and indefinite articles.  For example:     PULL (THE) LEVER TO OPERATE      CARS MAY BE TOWED WHEN (THEY ARE) PARKED ILLEGALLY      PLEASE USE (THE) OTHER DOOR   Is there a name for this custom? And what is the best way to describe it? Is there a list of Ubuntu versions with default corresponding Linux kernel versions somewhere?  I would specifically like to know the most recent version of Ubuntu that still used Linux Kernel 2.x. What are the limits of the Room of Requirement? Upgrading my Wubi 11.04 to 11.10 I want to upgrade my current Wubi installation (Ubuntu 11.04) to Ubuntu 11.10. Is this possible to do inside Wubi? Or do I have delete Wubi from Windows, therefore deleting my current Ubuntu installation; and then install 11.10 via the Wubi installer?  Thanks Do I need a transit visa in Zurich on the way to India? I am travelling from US to India, with an Indian passport but expired F1 visa and going there to get my H1B visa stamping. I have a stop at Zurich for 2 hours while flying to India. Do I need a transit visa for my trip? By accident, I messed up a configuration file. How do I recover the default version without reinstalling or affecting other things? Problem :   Solve in $\mathbb N$ :   $8n+9\equiv 0\pmod{1163}$  The mathematics give me : $n=435+1163k$ , $k\in\mathbb N$   $1163$ is a prime number   $8n\equiv -9\pmod{1163}$   $8n\equiv 1154\pmod{1163}$   I don't know any ideas to complete my work ?   I have to see your hints How to increase swap size? Can I be admitted into a PhD program with non-thesis MS degree? If I do a non-thesis MS program in statistics, will I be eligible for an admission in to a PhD program in Statistics later in my life? The MS program I am looking into is a non-thesis program but I still have to complete a research project requirement (which involves oral presentation as well as some form of a written report) towards the end of the program...I am still not sure if it can lead to a PhD program though.  Thanks!