Please prove: Let $f:(a,b) \to R$ be differentiable function, and let $c \in (a,b).  $  If $\lim_{x \to c}f'(x)$ exists and is finite, then this limit must be $f'(c)$.  I tried doing it directly but it doesn't quite work out. I experimented on some common functions (like $\sin(x), x^2$) and it seems pretty obvious.  I'm thinking about using contradiction, although after writing down some definitions, I couldn't proceed. Or, probably if I could show that $f'(x)$ is continuous, then I win. Any idea? Thanks!
Let $f$ be a real-valued function continuous on $[a,b]$ and differentiable on $(a,b)$. Suppose that $\lim_{x\rightarrow a}f'(x)$ exists. Then, prove that $f$ is differentiable at $a$ and $f'(a)=\lim_{x\rightarrow a}f'(x)$.    It seems like an easy example, but a little bit tricky. I'm not sure which theorems should be used in here.    ==============================================================  Using @David Mitra's advice and @Pete L. Clark's notes I tried to solve this proof. I want to know my proof is correct or not.  By MVT, for $h&gt;0$ and $c_h \in (a,a+h)$ $$\frac{f(a+h)-f(a)}{h}=f'(c_h)$$ and $\lim_{h \rightarrow 0^+}c_h=a$.    Then $$\lim_{h \rightarrow 0^+}\frac{f(a+h)-f(a)}{h}=\lim_{h \rightarrow 0^+}f'(c_h)=\lim_{h \rightarrow 0^+}f'(a)$$   But that's enough? I think I should show something more, but don't know what it is.
Detailed Bathymetric/Bathymetry contour data (0-200M) A smooth vector field on $S^2$ vanishing at a single point I need to find a vector field as described in the title.  I was given a couple hints, and this is what I have so far. Let $\varphi:S^2\setminus\{N\}\to\mathbb{R}^2$ be stereographic projection ($N$ is the north pole). Let $Y$ be a smooth, non-zero vector field on $\mathbb{R}^2$ that dies off to $0$ "at infinity." Since $\varphi$ is a diffeomorphism, we can push $Y$ forward along $\varphi^{-1}$ to get a smooth vector field $(\varphi^{-1})_*Y_{\varphi(p)}$ on $S^2$. But this vector field is not zero at the north pole... it's undefined. So can I just define my vector field on $S^2$, call it Z, to be the pushforward of $Y$ (as above) away from $N$ but $0$ at $N$? And if so, do we still have smoothness? How to use an item in my off-hand? Where are executables for programs stored in Ubuntu?  An application (Komodo Edit) is asking me to identify an application to be used as a web browser. I've become used to just entering the application name as a command for situations such as these, but this scenario got me thinking.   I know in Windows it would just be the relevant application folder in the 'program files' folder, but I'm assuming things are a bit different on Linux?  I thought somewhere like bin would be logical but this appears to standard Linux/Unix applications. Where would I find the binary executable for applications stored on my system? I have an Samsung Galaxy S3 Neo.  Bus 002 Device 042: ID 04e8:6860 Samsung Electronics Co., Ltd GT-I9100 Phone [Galaxy S II], GT-I9300 Phone [Galaxy S III], GT-P7500 [Galaxy Tab 10.1]   The keyboard &amp; language setting is português.  I today installed Android Device Manager and to test it's functionality I locked the phone with the password: "abcd" (4 letters were required).  After entering the password and confirming with the "OK" button of my keyboard, the phone showed me that the password is wrong. "Senha incorreta".  I reset the password after some tries to "123456" and the phone reloaded the lock screen. Again the password showed up to be wrong. I also tried the alphabetic password: "abcdefg" - same error.  I changed the language of my Google account from German to Portuguese and reset the password to "123456" again and I tried to really reboot my phone and unlock the SIM card again. But anyways I can not get through my lock of the Android Device Manager.  I tried to enter the password 35 times wrong with hope to get a chance to authenticate with my Google Account login.  I also tried to connect the phone with adb but it does not show up in this state with:  adb devices   So what can I do to get access to my phone again? Euler-Lagrange equation in curved spacetime: how to calculate $\partial \mathcal{L}/\partial(\nabla_\nu A_\mu)$? My problem is mostly with notation. When we apply the Euler-Lagrange equation to the Lagrangian scalar $$\mathcal{L}= -\frac{1}{4}F^{\mu\nu}F_{\mu\nu}$$ Where $$F_{\mu\nu} = \partial_\mu A_\nu -\partial_\nu A_\mu= \nabla_\mu A_\nu -\nabla_\nu A_\mu$$ and I start with $$\frac{\partial\mathcal{L}}{\partial A_\mu}=\nabla_\nu\left(\frac{\partial \mathcal{L}}{\partial(\nabla_\nu A_\mu)}\right).$$   Now I rewrite the lagrangian as $\mathcal{L} = -\frac{1}{4}g^{\mu\alpha}g^{ \nu\beta}(\nabla_\mu A_\nu-\nabla_\nu A_\mu)(\nabla_\alpha A_\beta-\nabla_\beta A_\alpha)$ $$\mathcal{L}= -\frac{1}{4}g^{\mu\alpha}g^{\nu\beta}\left[(\nabla_\mu A_\nu)(\nabla_\alpha A_\beta)-(\nabla_\mu A_\nu)(\nabla_\beta A_\alpha)-(\nabla_\nu A_\mu)(\nabla_\alpha A_\beta)+(\nabla_\nu A_\mu)(\nabla_\beta A_\alpha) \right]$$ My problem is calculating $\frac{\partial \mathcal{L}}{\partial(\nabla_\nu A_\mu)}$, how are the rules of partial derivatives generalised to covariant ones? Also, is $\frac{\partial\mathcal{L}}{\partial A_\mu}=0$, is that correct ? How to iterate over arguments in a Bash script I have a complex command that I'd like to make a shell/bash script of.  I can write it in terms of $1 easily:  foo $1 args -o $1.ext   I want to be able to pass multiple input names to the script. What's the right way to do it?    And, of course, I want to handle filenames with spaces in them. How to get the result of OnPostExecute() to main activity because AsyncTask is a separate class? The multi-part identifier could not be bound Motivation  I want to include a list of related equations, say, for a proof, in my LaTeX document. As far as I know, I have two good options, eqnarray and align.   Question  What is the difference between eqnarray and align, and how do I know which I should be using? Or does it matter at all? Unable to forward search Bash history similarly as with CTRL-r No group of order 400 is simple Show that if $a$ and $b$ are positive integers, then $$\left(a +\frac12\right)^n + \left(b+\frac{1}{2}\right)^n$$is an integer for only finitely many positive integers $n$.   I tried hard but nothing seems to work. :( How to work with "FIFO" in C# .NET? I'm making an app with a custom list in multiple languages.  Is there an easy way so that I only have to create the site columns, content type and list once? So I don't have to create them for each language?  My app will be in English, French and Dutch.  My lists elements.xml looks like this  &lt;?xml version="1.0" encoding="utf-8"?&gt; &lt;Elements xmlns="http://schemas.microsoft.com/sharepoint/"&gt;     &lt;!-- Do not change the value of the Name attribute below. If it does not match the folder name of the List project item, an error will occur when the project is run. --&gt;     &lt;ListTemplate         Name="List1"         Type="10000"         BaseType="0"         OnQuickLaunch="TRUE"         SecurityBits="11"         Sequence="410"         DisplayName="List1"         Description="My List Definition"         Image="/_layouts/15/images/itgen.png"/&gt;   &lt;Field       Name="Adres"       ID="{2a32d9eb-ad3b-4dbb-abba-66dd01ca7c6e}"       DisplayName="$Resources:Adres"       Type="Text"       Required="TRUE" /&gt;   &lt;Field       Name="Naampje"       ID="{ac47be45-2f43-4d16-9a06-162c85945ec9}"       DisplayName="$Resources:Naampje"       Type="Text"       Required="FALSE" /&gt;   &lt;Field     Name="Nickname"     ID="{6B0A2CD7-A7F9-41ca-B932-F3BEBB603793}"     DisplayName="$Resources:Bijnaam"     Type="Text"     Required="FALSE" /&gt;  &lt;/Elements&gt;   Well if i change my language in Sharepoint (Office 365) it's not changing the titles from dutch to the other languages i have added. A better answer was posted long after accepting another. Should I accept the new one? $ \lim_{n \to \infty} \left(\frac 1{n^2+1}+\frac 2{n^2+2}+\frac 3{n^2+3}+\cdots +\frac n{n^2+n}\right)$ An init-script does not get called on shutdown I have set up an init-script to control the state of a VirtualBox VM:  #!/bin/sh #chkconfig: 35 99 5 #description: vTiger virtual machine  ### BEGIN INIT INFO # Provides: vtigervm # Required-Start: $local_fs # Requider-Stop: $stop_fs # Default-Start: 3 5 # Default-Stop: 0 1 2 6 # Short-Description: Manage vTiger virtual machine # Description: Utility to start and stop vTiger virtual machine on VirtualBox ### END INIT INFO  start() {     echo -n "Starting vTiger"     echo     su myuser -c '/usr/bin/VBoxManage startvm "vTiger" --type headless'     echo "Started virtual machine" &gt;&gt; /var/log/messages }  stop() {     echo -n "Shutting vTiger down..."     echo     su myuser-c '/usr/bin/VBoxManage controlvm "vTiger" acpipowerbutton'     while [ ! -z "`su - juhani -c '/usr/bin/VBoxManage list runningvms | grep vTiger'`" ]; do         echo -n "."         sleep 1     done     echo "Done."     echo "Stopped virtual machine" &gt;&gt; /var/log/messages }  status() {     echo -n "Running VMs: "     su myuser -c '/usr/bin/VBoxManage list runningvms'     echo     if [ -z "`su - juhani -c '/usr/bin/VBoxManage list runningvms | grep vTiger'`" ]; then         RETVAL=3         else         RETVAL=0     fi     echo "Queried virtual machine status" &gt;&gt; /var/log/messages }  echo "Called virtual machine management script with: $1" &gt;&gt; /var/log/messages case "$1" in   start)         start    ;;    stop)         stop    ;;    restart|try-restart|condrestart|reload)         stop         start    ;;    status)         status    ;;    *)         echo $"Usage: $0 {start|stop|restart|status}"         exit 1    ;; esac  exit $RETVAL   The system is CentOS 6.5. If I manually run service [start|stop|status] vtigervm, it works as I expect it to. If the VM is running, $? after the service vtigervm status returns 0, and 3 if it is stopped. I installed it with chkconfig and it created among a few others rc5.d/S99vtigervm and rc0.d/K05vtigervm.  The problem  When I start the system it starts the "service", but on shutdown it does not even run the scripts.  grep "virtual machine" /var/log/messages shows:  *[machine starting]* Called virtual machine management script with: start Started virtual machine *[shutdown -h now]* *[machine stopped]*   What I expect:  *[machine starting]* Called virtual machine management script with: start Started virtual machine  *[shutdown -h now]* Called virtual machine management script with: status Queried virtual machine status Called virtual machine management script with: stop Stopped virtual machine *[machine stopped]*   File permissions &amp; etc:  # ls -lah /etc/rc0.d/ lrwxrwxrwx. 1 root root 13 7.2. 17:49 /etc/rc0.d/K05atd -&gt; ../initd.d/atd lrwxrwxrwx. 1 root root 18 9.2. 00:06 /etc/rc0.d/K05vtigervm -&gt; ../initd.d/vtigervm Why do some games allow offline Steam achievements? I know of  where it is asked if you can get Steam achievements offline. The accepted answer to that was no.   But I do earn achievements for some Steam games while playing in offline mode!  For example, right now I am earning achievements for Cave Story+ as I play in offline mode. In other games I also earn achievements while playing in offline mode. Not just when I am disconnected but still in "online" mode as might be suggested in the linked question.  But what is really bugging me is that this does not happen with all games that have Steam achievements.  So, is allowing offline achievements something that Steam allows, the game devs allow, or is it something decided by a magical elf in the steam servers rolling some d20 dice around?  Edit:  As per request, I was able to attain visual evidence:    You should be able to see the achievement-get in action while in offline mode.  What is worth noting, (and something that I forgot to mention before) is that I only started earning achievements after I had went into online mode with the game installed. This is why I was able to earn this achievement with the ones after unlocked. I had started the game for the first time in offline mode, and did not notice that I was not earning achievements. When I had connected once, I did not have to start the game while online. I was able to earn achievements offline.  I also must state publicly: I do not play on easy. This was for testing purposes only people. It was for Science! Moderncv package - cventry date width