I would like to find the negation of this statement of continuity of a function $f \colon \mathbb{R} \to \mathbb{R}$ at a point $y$:  $\forall \epsilon &gt; 0$, $\exists \delta &gt; 0$, such that $\forall x$ if |$x-y$| &lt; $\delta$ then |$f(x)-      f(y)$| &lt; $\epsilon$  So, when considering the negation of this statement I would want to consider negating the following components of the statement:  [$\forall \epsilon &gt; 0$] becomes [$\exists \epsilon &lt; 0$]? (I am a little confused because not($\forall x, P(x)$ becomes $\exists \epsilon$ such that not$(P(x))$, but would the statement that $\epsilon &gt; 0$ be considered the predicate P($\epsilon$) in this instance? If it is then I assume what I wrote above is correct then?)  [$\exists \delta &gt; 0$] becomes [$\forall \delta &lt; 0$] (similar reasoning as above)  [$\forall x$] becomes [$\exists x$] (here I am a little confused because I see the universal quantifier and want to negate it into the existential quantifier but the definition of the negation of the quantifiers includes a not statement of the predicate, but there is no predicate in this component. $x$ is used in a subsequent implication, which I am assuming is a separate negation on its own. Is this correct thinking?)  [if $|x-y|$ &lt; $\delta$ then $|f(x)-f(y)| &lt; \epsilon$] would become [$|x-y| &lt; \delta$ and $|f(x)-f(y)| &gt; \epsilon$]. Here I am just using the definition of the negation of the implication, which is $\text{not}(A \Rightarrow B) = (A \text{ and not}(B))$.   so that my negation of the statement of continuity here is:  $\exists \epsilon &lt; 0, \forall \delta &lt; 0$ such that $\exists x,  |x-y| &lt; \delta$ and $|f(x)-f(y)| &gt; \epsilon$  Looking at the statement as a whole it seems like the first two components should only have their quantifiers changed and not their inequalities, but the rest of the statement seems to make sense to me as describing a point of discontinuity  on the function. Is my reasoning here correct? Or am I missing something entirely. I appreciate any help with this problem.
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
I would like to get the multi pattern match with implicit AND between patterns, i.e. equivalent to running several greps in a sequence:  grep pattern1 | grep pattern2 | ...   So how to convert it to something like?  grep pattern1 &amp; pattern2 &amp; pattern3   I would like to use single grep because I am building arguments dynamically, so everything has to fit in one string. Using filter is system feature, not grep, so it is not an argument for it.    Don't confuse this question with:  grep "pattern1\|pattern2\|..."   This is an OR multi pattern match. What should I do, as my submitted paper is still under review after 1 year? Display solution/file path in the Visual Studio IDE I frequently work with multiple instances of Visual Studio, often working on different branches of the same solution.   used to display the full path of the current source file in its title bar, but Visual&nbsp;Studio 2005 doesn't appear to do this. This makes it slightly more awkward than it should be to work out which branch of the solution I'm currently looking at (the quickest way I know of is to hover over a tab so you get the source file's path as a tooltip).  Is there a way to get the full solution or file path into the title bar, or at least somewhere that's always visible, so I can quickly tell which branch is loaded into each instance? What can I do when getting “This question body does not meet our quality standards”? While trying to ask a question, I got the error message     This question body does not meet our quality standards. Please make sure that it completely describes your problem - including what you have already tried - and is written using proper grammar.    Why am I getting this message? What can I do to get the system to accept my question? Can you be more specific? What are good resources on how to ask high-quality questions?      For more information, see  in the . I have a Java bean like this:  class Person {   int age;   String name; }   I'd like to iterate over a collection of these beans in a JSP, showing each person in a HTML table row, and in the last row of the table I'd like to show the total of all the ages.  The code to generate the table rows will look something like this:  &lt;c:forEach var="person" items="${personList}"&gt;   &lt;tr&gt;&lt;td&gt;${person.name}&lt;td&gt;&lt;td&gt;${person.age}&lt;/td&gt;&lt;/tr&gt; &lt;/c:forEach&gt;   However, I'm struggling to find a way to calculate the age total that will be shown in the final row without resorting to scriptlet code, any suggestions? What is a good criterion or rule-of-thumb for determining when a control should be greyed-out and when should it be completely invisible? What's the easiest way to unrar a file? I just realized I can't unrar .rar files on my Ubuntu machine!  What's the easiest way to accomplish this fundamental task? If possible I'd love something like 7Zip that I can just right Click, and extract and have the program worry about how to do it regardless if it's a .zip, a .rar or a .foo compressed file. Prove that the multiplicative groups $\mathbb{R} - \{0\}$ and $\mathbb{C} - \{0\}$ are not isomorphic. Is my proof correct? I have made use of the fact isomorphism preserves order of elements, which I proved couple of exercises back. I am also interested in other ways of proving it. Is there a more explicit way or is this explicit enough?    Problem Prove that the multiplicative groups $\mathbb{R} - \{0\}$ and $\mathbb{C} - \{0\}$ are not isomorphic.  Solution Recall that isomorphism preserves order of elements and hence if there exists an isomorphism from $\phi: \mathbb{C}-\{0\} \mapsto \mathbb{R}-\{0\}$, then $x \in \mathbb{C} - \{0\}$, and $\phi(x) \in \mathbb{R} - \{0\}$, then $\vert x \vert = \vert \phi(x) \vert$. Now note that the element $i \in \mathbb{C} - \{0\}$ has order $4$. However, no element in $\mathbb{R}-\{0\}$ has order $4$. Hence, no isomorphism can exist. Hence, the multiplicative groups $\mathbb{R} - \{0\}$ and $\mathbb{C} - \{0\}$ are not isomorphic.    Thanks What is "my guy syndrome" and how do I handle it? Can I include the completion of Udacity and Coursera classes I have attended in an academic CV? Can I include in my academic CV that I have attended and completed the  and  classes I have been taking? I understand that these are not anything major, but the courses definitely gives a good overview and starting formal study on the subject becomes less cryptic. If I can include them without any negative impacts, then under what section should I include them? How can you tell when a sunset is going to be spectacular? I have a spot that I like to take sunset pictures. I find that the most spectacular moments last for only maybe 5-10 minutes. What I would like to do is know about a half hour before that a sunset is going to be spectacular, so I can grab my camera, tripod, walk 5-10 minutes to my sunset spot, set up, and have a few minutes to spare in case something happens. However, I've often missed really cool sunsets, because I just don't know when to photograph them. How can you tell beforehand when a sunset is going to be spectacular? Thanks! Hello [Comma?] John, I have a follow up question to this question:   My question is: when should you add a comma when starting a letter/email with     Hello John,      It was fun to see ...   I am pretty sure that a comma goes after the name, but lately I have seen several people (academics) write      Hello, John,       It was fun to see ...   with a comma after the Hello.  I am wondering when a comma there is appropriate.    I see this question  but I am wondering about the situation where you have a comma after the name also. Travel on a Schengen visa stating EEA family member without spouse? The company I work for makes hardware that communicates to the computer though a serial port. Third party companies write software that communicates with our hardware.   There are times when I need to diagnose our hardware. However, a third party software app connects to the serial port when Windows starts up, blocking any other connection. I don't know the name of this application/service and it's not always the same one.  Is there any way to either:   Find the name/pid of the app/service that is currently using a given serial port or Steal the serial port connection from another app.   vb.net preferably, but I'll take a language agnostic answer as well. It's not part of the artwork and doesn't seem to be hurting anything, but I get this on a lot of client logos and I'm wondering how to get rid of it.   I think it's part of the artboard, but can't find anything to select/change/remove it. The Maximum possible order for an element $S_n$ "Has/Have/Had" as a main verb and auxiliary verb I need help regarding the use of has/have/had both as a main verb and as an auxiliary verb.  In the sentence below, the word has acts as the main verb. I would normally follow it with the phrase "so do I."        Kenny has a car, and so do I.      On the other hand, the word has in the following sentence acts as an auxiliary verb. I would normally follow it with the phrase "so have I."        Kenny has bought a car, and so have I.      a) Are sentences 1 &amp; 2 above correct? I'm under the impression that so do I is used when has/have/had is the main verb of a sentence, and so have/had I if it's an auxiliary verb in a sentence.  b) Does the verb has/have always need a supporting "auxiliary verb" (either implied or mentioned)? (e.g., Kenny has a car = Kenny does have a car.) Are global variables bad? Identify this YA book series about dragons, acid blood, and dangerously cold nights I'm trying to remember the name of a series of books about a boy who works with dragons.  Dragon's blood (or maybe it was slobber?) is acidic, and the children who wrangle these dragons often end up with deep pock marks.  On this world, days are like ours but nights get very cold, requiring anyone stuck outside to find a shelter until dawn (not the false dawn)  The other thing I remember is that in one of the books by slicing open a dragon and spending the night, the character emerged with special dragon powers. Symbols instead of numbers as footnote markers How can I make LaTeX use symbols (*, †, ‡, and so on) instead of numbers to mark footnotes?  (The numbers are confusing because I use superscripted numbers for citations.)