Prove that $\forall x \in \mathbb{R}, f(x)=0$
Prove $f $ is identically zero if $f(0)=0$ and $|f'(x)|\le|f(x)|$
"Unanswered" question sort order on profile pages To help users keep track of (and hopefully get answers to) unanswered questions, it would be useful to have an "unanswered" sort order on a user's profile page.  It would order the list of questions with unanswered questions first.   This would be in addition to the "recent," "views," "newest," and "votes" sort orders. Evaluating $\int_{0}^{1} dx\frac{\log(1+x)}{1 + x^2}$ Possible Duplicate:          $$\int_{0}^{1} dx\frac{\log(1+x)}{1 + x^2}$$  I am having a hard time deriving the answer, $\frac{\pi}{8} \log(2) $.  I have tried Taylor expansion of both numerator and denominator, both seem too complicated and fruitless. Windows 7 SP1 Windows Update stuck checking for updates Let $G$ be a finite semi-group with identity such that it has only one idempotent.Is $G$ a group?  It only remains to show that for any $a\in G$ $\exists b\in G$ such that $ab=ba=e$ where $e$ is the identity of $G$  Also $e$ is the only idempotent of $G$ .How to proceed next? How can I set up password-less SSH login? The relation between acceleration and inertial force Lost email settings and no icon for Mail I've lost my Settings → Mail, Contacts, and Calendars on my iPhone 5.  No icon for Mail and no Mail settings.  How do I get it back? How can I listen to communication between a process and a serial port? On a Gentoo Linux machine, a process is communicating with another computer via serial port (/dev/ttyS1). I want to listen to this communication (i.e. what is being sent/received by the process) without further interfering.   How can I achieve this with just basic Linux tools? I even can change the serial port the process is communicating with, so maybe I can 'shortcut' two devices (/dev/ttyS1 and /dev/xxx) listen to all communication, when all output to /dev/ttyS1 is forwarded to /dev/xxx and vice versa. But how do I do it? Something with socat? Getting hybrid graphics to work nvidia-prime GT650M I recently made the switch to Ubuntu. Everything on my laptop works fine except my hybrid graphics card. It is Intel graphics combined with Nvidia GeForce GT650M. I tried Bumblebee, but primusrun doesn't work (segmentation fault, core dumped) and optirun works with glxspheres64, but steam games look worse with optirun than with Intel graphics (e.g. Surgeon Simulator 2013 runs fine with Intel and hangs a lot with optirun)   I tried Nvidia Prime, but it gives me a black screen on startup after install, so I can't do anything and have to power off my PC with the power button. I also tried nvidia-prime 0.5 on Ubuntu 14.04, and then I get 'low-graphics mode' and I can't get to Unity. Calculate limiting distribution of $\frac{\sum_{i=1}^n X_i}{\sum_{i=1}^n Y_i}$ Where does gravity get its energy from? Assuming i have a line that i want to add to a file without opening an editor.  How could i append this line  alias list='ls -cl --group-directories-first'   to this file  config.fish I wonder which of the two is correct:        I have a red and a blue car.   I have a red and a blue cars.      or        Everyone has a left and a right arms.   Everyone has a left and a right arm.      My (non-native) inclination would be for option 1, as it sounds better.  On the other hand, in the following examples, the plural sounds better to me.    Everyone has left and right arms. I have old and new cars. (when I only have only one old car and one new car) The left and the right arms are...   But are they correct? Deleting array elements in JavaScript - delete vs splice How to recenter an object's origin? So, I'm working creating a lamp... yes, newbie here.  After modeling for a bit, I noticed that my point of origin was no longer the center of my object:    How to I reset an object origin point to the center of it's geometry? If two gravitational waves came in contact with each other what would happen? In another question entirely, what happens when a higher gravitational field interacts with a weaker one. Password protect folder without disk images? Is this possible? I know you can protect a .dmg file, but is there a more convenient solution, by using a third party application? So when I got Steam working offline I had 26 games listed in my library but only 2 installed. Recently I brought a bunch more so I want to update my library on my Offline PC with the full list that's on my Online Laptop.  I thought that all I would need just copy the data from userdata/[ID]/config/localconfig.vdf however when I do the number goes down to 2 (the number of games I do have installed) and also the OST I got with Monster Monpiece doesn't appear. When I restore just that file the list and the music listing returns.  So what do I need to copy over from my Online Laptop to the Offline PC to update the list of games in the library? A friend of mine asked me the following question:      Suppose you have a basket in which there is a coin. The coin is marked   with the number one. At noon less one minute, someone takes the coin   number one and put into the basket two coins: the number two and the   number three. At noon less $\frac{1}{2}$ minute, he takes the coin   number two and put into the basket the coins number $4$ and the number   $5$ and so on. At noon less $\frac{1}{k}$ he takes the coin number $k$   and put two other coins.       The question is: How many coins I can find in the basket at noon?   Intuitively the answer is $\infty$ because I added one coin for infinite times. But the correct answer seems to be $0$ because every coin numbered $k$ has been removed at noon less $\frac{1}{k}$ of minute. What is the correct answer? Thanks. Kernel of a substitution map