Let $f: \mathbb{R}\rightarrow{\mathbb{R}}$ be given by $f(x)=sen(\frac{1}{x})$ si $x\neq 0$ and $f(x)=c$ si $x=0$, where $c\in [-1,1]$  For which values of c there is an antiderivative?  Proof (attempt): I want to use a proposition: $f$ is continuous if, and only if, $osc(\Omega)=0$, because $f$ is discontinuous at all points $c\in [-1,1]$.
Let $ f: \mathbb {R} \rightarrow {\mathbb {R}} $ be given by   $$ f (x) = \left \{\begin {matrix} \sin ({\frac {1} {x}}), &amp; \text {if} \phantom {a} x \neq 0; \\  c, &amp; \text {if} \phantom {a} x = 0 \end {matrix} \right. $$ where $ c \in [-1,1]$. For what values ​​of $c$ is there an antiderivative of $f$?  I do not know of a theorem of the form "if and only if" that  it tells me when a function $ f $ has no antiderivative.  Anyone have a suggestion?
How to change boot priority? Can we please move the  "Start a bounty" link to be in the same location as all of the other action buttons? It makes no sense to me for this to be buried after the comments - It's an action on the Question. It can be very difficult to locate, especially when your question gets .  I don't think this should be that difficult to implement - it's just moving the location of a design element. Can we please get this button moved? Find all subgroups of $\mathbb{Z}\times\mathbb{Z}$. Find all subgroups of $\mathbb{Z}\times\mathbb{Z}$.   I can find all the infinite subgroups of the form $n\mathbb{Z}\times m\mathbb{Z}$, where $n,m$ run over $\mathbb{Z}$. But I don't know how to write out all the finite subgroups in a uniform expression.  Any suggestions? Thanks in advance. ElasticSearch doesn't start after installing 5.6.4 I have a multiboot system set up. The system has three drives. Multiboot is configured with Windows XP, Windows 7, and Ubuntu - all on the first drive. I had a lot of unpartitioned space left on the drive and was reserving it for adding other OSes and for storing files there in the future.  One day I went ahead and downloaded Partition Wizard and created a logical NTFS partition from within Windows 7, still some unpartitioned space left over. Everything worked fine, until I rebooted the computer a few days later.  Now I'm getting:    error: unknown filesystem.   grub rescue&gt;   First of all I was surprised not to find any kind of help command, by trying:    help, ?, man, --help, -h, bash, cmd, etc.  Now I'm stuck with non-bootable system. I have started researching the issue and finding that people usually recommend to boot to a Live CD and fix the issue from there. Is there a way to fix this issue from within grub rescue without the need for Live CD?  UPDATE  By following the steps from , I was able to boot to initramfs prompt. But not anywhere further than that.  So far from reading the manual on , I was able to see my drives and partitions using ls command. For the first hard drive I see the following: (hd0) (hd0,msdos6) (hd0,msdos5) (hd0,msdos2) (hd0,msdos1)  I now know that (hd0,msdos6) contains Linux on it, since ls (hd0,msdos6)/ lists directories. Others will give "error: unknown filesystem."  UPDATE 2  After the following commands I am now getting to the boot menu and can boot into Windows 7 and Ubuntu, but upon reboot I have to repeat these steps.    ls ls (hd0,msdos6)/ set root=(hd0,msdos6) ls / set prefix=(hd0,msdos6)/boot/grub insmod /boot/grub/linux.mod normal   UPDATE 3  Thanks Shashank Singh, with your instructions I have simplified my steps to the following. I have learned from you that I can replace msdos6 with just a 6 and that I can just do insmod normal instead of insmod /boot/grub/linux.mod. Now I just need to figure out how to save this settings from within grub itself, without booting into any OS.  set root=(hd0,6) set prefix=(hd0,6)/boot/grub insmod normal normal   UPDATE 4  Well, it seems like it is a requirement to boot into Linux. After booting into Ubuntu I have performed the following steps described in the :  sudo update-grub sudo grub-install /dev/sda   This did not resolve the issue. I still get the grub rescue prompt. What do I need to do to permanently fix it?  I have also learned that drive numbers as in hd0 need to be translated to drive letters as in /dev/sda for some commands. hd1 would be sdb, hd2 would be sdc, and so on. Partitions listed in grub as (hd0,msdos6) would be translated to /dev/sda6.  UPDATE 5  I could not figure out why the following did not fix grub:  sudo update-grub sudo grub-install /dev/sda   So I downloaded  based on an answer from  post. That seemed to do the trick after I picked the "Recommended Repair (repairs most frequent problems)" option. Disable Tabbing in Firefox 3, Chrome 4 and IE8? Tabbed Browsing may be nice for reading news sites in one sitting, but I still think they are a usability nightmare if you need to keep the page open for a long time, e.g., for testing it while developing or as a reference for something.  The nice thing about the Windows Taskbar (before Windows 7 ) is that it's easy and natural to remember where things are. I can have 30 tabs in it, but because I know what each one is, I know where I have to click to open something - it's the most efficient way of working I've encountered so far.  Sadly, Tabbed Browsing completely breaks this efficiency because it is not possible to specifically click on a tab in the taskbar, I have to click on the browser, search the tab and then click it, which means at least 2 clicks.  Long story, short question: Is there an extension or setting to completely disable tabbing on Firefox 3, Chrome 4 and Internet Explorer 8 and have one Window/Taskbar entry for each open page? I know that in Chrome I can just drag a tab out of a Window to create a new one, but if there is a general way, that would be easier.  Edit: IE8 conveniently offers an option for that: Tools - Options - Tabs - Enable Tabbed Browsing. Article "communicated by" What does it mean when it says under a journal article that it has been communicated by "XYZ" where XYZ is not the author but some other scholar with a very strong reputation? What is the relationship to the actual author and/or the content? Is this some sort of seal of approval to get results out and known quickly? (I am specifically wondering in the context of mathematics and mathematical physics.) Can you use your off-handed weapon for your main attack action? My friend is 2 weapon fighting a flesh golem. He has one sword that does lightning damage, and one sword that does fire damage. His right hand holds the lightning sword, and his left hand holds the fire sword.  My friend wants to attack with his left hand because lightning heals the golem. However, the DM says that because his left hand is his left hand, he can't use it as his "main" hand, and only attack with it. Is this rules as written/intended? Count and limit the number of files uploaded (HTML file input) Proving that ${x +y+n- 1 \choose n}= \sum_{k=0}^n{x+n-k-1 \choose n-k}{y+k-1 \choose k} $ How can I prove that $${x +y+n- 1 \choose n}= \sum_{k=0}^n{x+n-k-1 \choose n-k}{y+k-1 \choose k} $$  I tried the following:  We use the :  $$y^{\underline k}=\underbrace{y(y-1)(y-2)\ldots(y-k+1)}_{k\text{ factors}},$$ so that $\binom{y}k=\frac{y^{\underline k}}{k!} .$   Then  $${x +y+n- 1 \choose n} = \frac{(x +y+n- 1)!}{n! ((x +y+n- 1) - n)!} =  \frac{1}{n!}. (x +y+n \color{#f00}{-1})^{\underline n} $$  And  $$ {x+n-k-1 \choose n-k}{y+k-1 \choose k}$$ $$\frac{1}{(n-k)!}.(x+n-k-1)^{\underline{n-k}}.\frac{1}{k!}.(y+k-1)^{\underline{k}}$$ $$\frac{1}{k!.(n-k)!}.(x+n-k-1)^{\underline{n-k}}.(y+k-1)^{\underline{k}}$$ $$\sum_{k=0}^n{n \choose k}(x+n-k-1)^{\underline{n-k}}.(y+k-1)^{\underline{k}}$$  According to the Binomial-coefficients:  $$ ((x+n-k-1) + (y+k-1))^{\underline{n}}$$ $$ (x+y+n\color{#f00}{- 2})^{\underline{n}}$$  What is wrong ? und How can I continue?  :/ Merging arrays with the same keys Light emission from the vicinity of a black hole The photograph of the black hole, at the centre of galaxy M87, is now a part of human history. We are told that the red-colour is caused by hot gas, matter &amp; radiation around the "edge", the Event Horizon. Given the power of the gravitational forces emanating from black hole, why aren't these three things drawn into the abyss? If light is bent/ drawn into the BH, how does this red image arise? The logic, here, would be that a body, on the Event Horizon, is immune from the gravity--which sounds pretty fatuous? I have my Xbox One set up to automatically download and install updates, the rest of my settings are shown below:    Is there a reason my games will show an update symbol on their tile, but they still don't download the update until I start the game manually? I am trying to run an executable file from a network drive on Windows 7 Professional (64-bit) in order to install a program, but, each time I try, it doesn't run and the following message appears:  {path-name} The specified path does not exist. Check the path, and then try again.   I am definitely connected to the network drive as I can see all of the directories and files, and can even open non-executable file types. Also, I am fairly confident that this is neither a firewall nor a permissions issue. What is the longest word you can come up with that the letters are all in alphabetical order? It's Friday again, how about some fun to get us into the weekend?  What is the longest word you can come up with for which all the letters in that word are in alphabetical order?   Rules:   English words only    Can't be a name of a place, person or other proper noun.   if it contains the same letter twice in a row, that does not disqualify it.   No fair looking it up on Google!   Update: One more rule to help you guys out.   The word can be in either ascending or descending alphabetical order. A required field that isn't really required? ...the email field in my clients data entry system is required but users of the system are allowed to enter na@na.com if the customer refuses or doesn't have an email address. Is this best practice? Have any suggestions or workarounds for their solution? Is it possible to highlight text inside of a textarea using javascript? Either changing the background of just a portion of the text area or making a portion of the text selected? I understand that for resistors in series we need to add R for each resistor to get the equivalent resistance, where R is resistance. I also understand the mathematical manipulations that show that for resistors in parallel we need to add the 1/R for each resistor to get the equivalent resistance. However I have a couple of questions that have to do with the physical interpretation of the mathematical manipulations involved.   First, I am trying to understand why in the case of parallel resistors we have to add 1/R's instead of adding R's as in the case of resistors in series i.e. could it be thought that the current in the wire is like a sort of water flow in the pipe that has two horizontal parallel pipes positioned one on top of the other and in between connected by a number of vertical pipes that have a tap each that could be adjusted to represent different resistances. Then at the top of each vertical pipe the water pressure P exists and at the bottom the pressure will be reduced by P*R.   For example, suppose we have three vertical pipes, so then the pressure at the bottom of each (after the tap) is P*R for the total of 3*P*R so why is the equivalent resistance not 3*R*P but 1/R = (1/R1 + 1/R2 + 1/R3) and how can I be able to visualize this relation? Downcasting in Java Upcasting is allowed in Java, however downcasting gives a compile error.   The compile error can be removed by adding a cast but would anyway break at the runtime.   In this case why Java allows downcasting if it cannot be executed at the runtime? Is there any practical use for this concept?   public class demo {   public static void main(String a[]) {       B b = (B) new A(); // compiles with the cast,                           // but runtime exception - java.lang.ClassCastException   } }  class A {   public void draw() {     System.out.println("1");   }    public void draw1() {     System.out.println("2");   } }  class B extends A {   public void draw() {     System.out.println("3");   }   public void draw2() {     System.out.println("4");   } } Why won't Mathematica display an answer?