Here's the question:     Let $f(x)=\sin \left( \frac{1}{x} \right)$ for $x\ne 0$ and let   $f(0)=0$. Show that $f$ is discontinuous on $\mathbb{R}$ and still has   the intermediate value property on $\mathbb{R}$.   It's easy to show that $f$ is discontinuous at $x=0$. Now, I go on to prove that $f$ has the intermediate value property. Let $a,b \in \mathbb{R}$ with $a&lt;b$. Assume that $y$ be some real such that $f(a) &lt; y &lt; f(b)$. If $y=0$ we are done for $f(0)=y$. If $y\ne 0$, I was hoping to use the continuity of sine on $\mathbb{R}$ but nothing seems work out.  Hints would be appreciated. I do not need full solution.
Let $$f(x)=\begin{cases} \sin \tfrac 1 x &amp;\text{if $x\ne 0$}\\ 0 &amp; \text{if $x = 0$.}\end{cases}$$ I have to show that $f$ has the intermediate value property. That is, for any $a &lt; b$, if $y$ is any real number such that $f(a) &lt; y&lt; f(b)$ or $f(a)&gt;y&gt; f(b)$, then there exists a $c \in (a,b)$ such that $f(c)=y$.  I feel like I kind of know how to go about completing this. I just am curious as to if I have to create a bound such as letting $a = -1$ and $b = 1$, or keep $a$ and $b$ both arbitrary.
SQL - Stored Procedure with Select Statement using IN (@Variable_CommaDelimitedListOfIDS) How do I redirect a domain to another domain locally? My /etc/hosts file looks like this:  127.0.0.1       localhost 10.20.7.67      testsitealpha.dev othersite.dev   testsitebeta.dev   The first two work. The last one doesn't. Why not? How do I make it work? The reason I am doing this, is because I have a test server (othersite.com), which is on the local network, but it's ip can vary. e.g. 10.20.7.98 one day, 10.20.7.35 another, etc.  So, how do I make my system always resolve testsitebeta.dev to the same ip as othersite.dev? Changing local IP address with networksetup command on OSX? My computer routinely receives connection from another host in the local network. However, the router changes its ip address quite frequently, and maybe netsh's equivalent, networksetup, can help? This should be a very easy question but it might just be that I'm confusing myself. So we have the definition of a function $f$ on $S$ being continuous at $x_0$:     For any $\epsilon$>0, there exists $\delta&gt;0$ such that: whenever $|x-x_0|&lt;\delta$, we have $|f(x)-f(x_0)|&lt;\epsilon$   And I assume the negation is     There exists $\epsilon$>0 such that for all $\delta&gt;0$, $|x-x_0|&lt;\delta$ yet $|f(x)-f(x_0)|\ge \epsilon$.   Now I want to show that the function $f(x)=\sin(\frac{1}{x})$ together with $f(0)=0$ cannot be made into a continuous function at $x=0$. So I need to show that there exists $\epsilon&gt;0$ such that for all $\delta&gt;0$, $|x|&lt;\delta$ yet $|f(x)|\ge\epsilon$.  Let $\epsilon = \frac{1}{2}$. Then no matter what $\delta$ we choose, let $|x|&lt;\frac{1}{2}$. It is certainly possible that $|f(x)|\ge \frac{1}{2}$, because, well, $\frac{1}{x}$ can really take on arbitrarily large value as $x$ is small.  Now, what confuses me is that, as $x$ gets small, $f(x)$ can certainly be greater than $\frac{1}{2}$ for infinitely many times, but it will be less than that infinitely many times, too. But I suppose it doesn't really matter. So I think there's something wrong with my negation but I couldn't figure out where.  Update: The correct version can be found . Watch for Lemma 4.6 Securing Cable when installing old work box I am installing an electrical outlet above a gas fireplace for TV.  I understand that electrical code requires securing the NM cable within 12" of outlet box.  I am curious how one goes about achieving this in a finished wall being that there is really no room to get hand, hammer, drill or screwdriver inside the wall through that box opening.   I am sure this is straightforward for experienced electricians but I have no idea how to go about this. How do I check if a package is installed on my server? I recently installed Ubuntu Server edition 13.10 (Saucy Salamander). But I have run into a bit of problem. At the end of the installation, it asks about several extra packages whether you want them or not, like OpenSSH server and virtual machine host. Here's what I'm looking at:  Now, my problem is that I can't remember if I have checked the &quot;virtual machine host&quot; checkbox. How do I check if my server has the corresponding packages? What packages come with the virtual machine host? If $x_n \to 0$, show that $\sqrt{x_n} \to 0$ Let $x_n \geq 0$ for all $n \in N$. If $(x_n) \to 0$, show that $(\sqrt{x_n}) \to 0$  Unless I am missing something, this seems simple to me. Since $x_n$ converges for an arbitrary $\epsilon^2&gt;0$, we have $|x_n| &lt;\epsilon^2 $ for $n\geq N$. Taking square roots here we have $\sqrt{x_n}&lt;\epsilon$ for all $n\geq N$ which is what we needed to show. Is that all? If so, then why $x_n \geq 0$ is important? Can Reduce *really* not solve for x here? Why is the area under a curve the integral? I understand how derivatives work based on the definition, and the fact that my professor explained it step by step until the point where I can derive it myself.  However when it comes to the area under a curve for some reason when you break it up into an infinite amount of rectangles, magically it turns into the anti-derivative. Can someone explain why that is the definition of the integral and how Newton figured this out? I have a question about my choice of name for publication.  I recently got married and legally changed my name to my husband's surname (for example's sake: from Hunter to Jones). So, despite legally being Ms Jones now, I would quite like to use the hyphenated name Hunter-Jones in the professional context and in publications. This is so that people realise that it is still me and so that I can somewhat keep my maiden name.  Is this possible or do I have to use my legal name professionally and when publishing? Thank you for any help Although Ubuntu 16.04 release notes mention that:  The default VIM package has been built against python3 instead of python2.  Yet after I installed vim with the following command:  sudo apt-get install vim   And checked:  vim --version   It showed:  -python -python3   Is it a bug? How do I get vim with Python support? I've previously used the default Email app on my CyanogenMod 7 EVO because I had an Exchange account at work. However, I switched jobs recently and my new place just uses GMail corporate accounts (which I enjoy). Since I don't have the Exchange account anymore, I'd like to move over to the GMail app because I prefer the UI and whatnot, but I can't for the life of me find any kind of "Select All" functionality in it.  The Email app allows me to do this from the options menu (Menu-&gt;Select All selects everything in the current folder). In GMail I noticed that if I select one or more items, then "Deselect All" becomes available in the options menu, but not "Select All". I've also tried various forms of long-pressing on messages, accounts, and folders to no avail.  Is there any way to do this in the GMail app, or am I just stuck without it? I searched the issue tracker and didn't find anything relevant, which surprised me a bit, to be honest. Obviously I can do this from a computer if I need to empty a folder/label, but I'd kind of like to retain the functionality on my phone, too. Am I missing some non-obvious (or perhaps even obvious) menu item somewhere? Can markdown mini replace an en dash with a normal hyphen in URLs? It looks as if the 10k-tools flagged post list is b0rken.  Next to the review link in the toolbar there is a flag counter showing there are 10 flags to be reviewed:    Clicking on that flag counter leads to an empty page:        The moment I visit that page (through clicking on the flags counter, or by going to the review queue first then clicking on Tools, selecting the flags tab if not already selected), the flags counter disappears from the toolbar:    but after a while (half an hour, an hour?) the flag count re-appears in the toolbar.  Either the counter is wrong (presumably taken from a Redis cache somewhere) or the 10k flag review page is falling over every time someone tries to load it.  This at first appeared to happen only on Meta, on Stack Overflow the flags page has plenty to review. However, there are now  on Stack Overflow. Ubuntu 20.04 still supporting Python2 Tensor product of a sheaf of module and a residue field Let $S$ be a scheme over a finite field $\mathbb{F}_q$ and $\mathcal{L}$ an invertible sheaf on $S$.   In Pink's article (, see section 1.4 on page 13), he states that:  For any section $\ell\in \mathcal{L}(S)$ and any point $s\in S$ we let $\ell(s)\in \mathcal{L}\otimes_{\mathcal{O}_S}k(s)$ denote the value of $\ell$ over the residue field $k(s)$ of $s$.  My question is: what is the definition of $\mathcal{L}\otimes_{\mathcal{O}_S}k(s)$?  We know we can define the tensor product of two $A$-modules or of two $\mathcal{O}_X$-modules, but what is the tensor product of an $\mathcal{O}_X$-module and a field? Why can't photons have a mass? Why can't photons have a mass? Could you explain this to me in a short and mathematical way? Why is the determinant the volume of a parallelepiped in any dimensions? For $n = 2$, I can visualize that the determinant $n \times n$ matrix is the area of the parallelograms by actually calculating the area by coordinates.  But how can one easily realize that it is true for any dimensions? I have TeX Live 2013 installed via MacTeX on my Snow Leopard laptop.  I copied the following code from page 247 of version 2.10 of pgfmanual:  \documentclass[a4paper,12pt]{report} \usepackage{tikz} \usetikzlibrary{decorations}  \begin{document}  \begin{tikzpicture} \node (a) {A}; \node (b) at (2,1) {B}; \draw (a) -- (b); \draw[decorate,decoration=brace] (a) -- (b); \draw[decorate,decoration={brace,mirror},red] (a) -- (b); \draw[decorate,decoration={brace,mirror,raise=5pt},blue] (a) -- (b); \end{tikzpicture}  \end{document}   Yet somehow I get this error:  ./Untitled2.tex:11: Package pgfkeys Error: I do not know the key '/pgf/decorati on/brace' and I am going to ignore it. Perhaps you misspelled it.  See the pgfkeys package documentation for explanation. Type  H &lt;return&gt;  for immediate help.  ...                                                l.11 \draw[decorate,decoration=brace]                                       (a) -- (b); ?    Is this a bug? What does "apt-get install !!" do and how to make it work? I have seen someone typing  apt-get install !!   He did that a few times and every time, a different program was installed. It seems like it automatically figured out what is needed next.  However, when I tried it on my Ubuntu 14.04.1 LTS virtual machine, it just showed some usage instructions. Googling for apt-get install !! doesn't seem to work very well. Many search results do not contain the double exclamation mark.  What does apt-get install !! do and what could be missing on my machine that it did not work?