could someone please explain to me the following question:  Let $f,g$ be continuous functions from $\mathbb{R}$ to $\mathbb{R}$ and suppose that $f(r) = g(r)$ for all $r \in \mathbb{Q}$. Is it true that $f(x) = g(x)$ for all $x \in \mathbb{R}$?  My claim is that it should be true since $f$ and $g$ are continuous from $\mathbb{R}$ to $\mathbb{R}$, so if $f(x) \neq g(x)$ for all $x \in \mathbb{R}$ then it cant be continuous on $\mathbb{R}$?   So, is my claim correct and if it is how am I supposed to prove it (please give me some hints) Also, if my claim is wrong, could you please explain to me as to why its wrong`
Akhil showed that the  is the same as the continuum, using as a step the observation that continuous functions that agree at rational points must agree everywhere, since the rationals are dense in the reals.  This isn't an obvious step, so why is it true?
Where is log file from rc.local? I have some commands to in my rc.local. I know that they are failing. How can I get log file with messages produced by executing rc.local? Where is it located?  I have checked the /var/log/boot.log. I know my messages are not there because I know already what is the reason of failure. But I still want to make sure from log file.  Note, I don't want to run script again, I could but I don't want. I would rather analyse wht happened during startup.  Thanks for any help.  Ubuntu 12.04 Desktop (if it matters) Mail App keeps popping up in the background in Mac OS Mojave Does a percentage quantity take singular or plural verb agreement? Does a percentage require a singular or plural verb, for example, do we say ten percent "go" or "goes"? Story whose final line is "Let there be light!" I read this series around 5-10 years ago. There were two parallel universes, ours and the universe of demons, and much of the series revolved around attacks in which demons created gateways to cross into our universe to kill people and wreak havoc, and had to be stopped by human magicians. Three youngsters (one guy from modern times, one from a few decades ago, and a girl from ancient Celtic times, all with funny names) end up being the key to the war when it's revealed that they are the three components of a mythical superweapon. These three are the only point-of-view characters in the series, each book being told from the PoV of just one of them. At the very end, they   destroy our entire universe so that, after killing all the demon masters, they can rebuild it and make sure everything happens as before up until the very last moment.  As they begin recreating the universe, the current PoV character (the one from modern times) says something like the following, the very last lines of the series. (sic on the quotation marks)  &quot;In the beginning, there was nothing but darkness. Then [name] said, 'Let there be light!'&quot; And there was light. Coolio! What does 'exec {fd}</dev/watchdog' do in Bash That is literal, {fd} isn't a placeholder. I have a script that does this, and does not source in anything, nor does it reference {fd} anywhere else.  Is this valid bash?  exec {fd}&lt;/dev/watchdog Restoring Windows XP PATH variable I installed an application that screwed my PATH variable. Now I cannot run anything from the command prompt. Is there any way I can restore PATH variable? How to determine whether a given Linux is 32 bit or 64 bit? When I type uname -a, it gives the following output.  Linux mars 2.6.9-67.0.15.ELsmp #1 SMP Tue Apr 22 13:50:33 EDT 2008 i686 i686 i386 GNU/Linux   How can I know from this that the given OS is 32 or 64 bit?  This is useful when writing configure scripts, for example: what architecture am I building for? How to interpret busybox "top" output? I am using BusyBox on a small embedded ARM system.  I'm trying to read the "top" output, in particular for the Python process listed.  How much real memory is this process using?  Also what does VSZ stand for?  The system only has 64MB of RAM.   Mem: 41444K used, 20572K free, 0K shrd, 0K buff, 18728K cached CPU:   3% usr   3% sys   0% nic  92% idle   0% io   0% irq   0% sirq Load average: 0.00 0.04 0.05 1/112 31667   PID  PPID USER     STAT   VSZ %VSZ %CPU COMMAND   777   775 python   S     146m 241%   3% /usr/bin/python -u -- dpdsrv.py Shell Script for logging into a ssh server I tried writing a shell script which can do automatic login into a ssh server using password which is mentioned in the script. I have written the following code:  set timeout 30 /usr/bin/ssh -p 8484 root@172.31.72.103 expect  {    "root@172.31.72.103's password"     {       send "password\r"     } }   This code is not running properly, still it is asking for the password. Can somebody please help me in solving this I have created some search entries in Google Chrome using 'Edit search engines'.  How can I share some of these entries with my colleagues? Definition of the Pauli group and the Clifford group There seem to be two definitions of the Pauli group. In Nielsen and Chuang, the Pauli group on 1 qubit is defined as \begin{align*} \mathcal{P}_1 = \{\pm I, \pm iI, \pm X, \pm iX, \pm Y, \pm iY, \pm Z, \pm iZ\} = \langle X, Y, Z\rangle. \end{align*} But in other references, such as ,  \begin{align*} \mathcal{P}_1 = \{\pm I, \pm X, \pm iY, \pm Z\} = \langle X, Z\rangle. \end{align*} The second definition makes more sense in the stabilizer formalism, since we usually only consider the action of Pauli strings of $X$ and $Z$. What bothers me is that by the second definition, the phase gate $S$, which is an element of the Clifford group, results in \begin{align*} SXS^\dagger = Y \notin \mathcal{P}_1. \end{align*} I know that the prefactor $\pm i$ somehow doesn't matter, but can someone justify it from a more mathematical/group-theoretical perspective? I've not seen/read any female Orcs mentioned in the movies or in the books. In one of the movies, an Uruk-hai is momentarily shown emerging from a giant, grown-Orc sized flesh packet. So are they created with spells, or do they follow the Standard Reproductive Model. This question pertains to L1 &amp; L2 regularization parameters in Light GBM. As per official documentation:  reg_alpha (float, optional (default=0.)) – L1 regularization term on weights.  reg_lambda (float, optional (default=0.)) – L2 regularization term on weights  I have seen data scientists using both of these parameters at the same time, ideally either you use L1 or L2 not both together.   While reading about tuning LGBM parameters I cam across one such case: Kaggle official GBDT Specification and Optimization Workshop in Paris where Instructors are ML experts. And these experts have used positive values of both L1 &amp; L2 params in LGBM model. Link below (Ctrl+F 'search_spaces' to directly reach parameter grid in this long kernel)      I have seen same in XGBoost implementations.  My question is why use both at the same time in LGBM/XGBoost.  Thanks. How do I upgrade from 14.04 LTS to 16.04.1 LTS? I've recently flagged the following , which I got in the flag queue.   The answer does not seem to attempt to address the question at all. It looks like a comment on one of the other answers (probably Luca's).  It got rejected with the following reason:  declined - flags should not be used to indicate technical inaccuracies,            or an altogether wrong answer   I did not flag this because I thought it was wrong, I flagged it because I didn't think it was an attempt to answer the question. For what it's worth, I don't get what the guy who provided that answer even tried to say.  Was I wrong to flag it? If so why? I couldn’t find any proof on the internet. I would appreciate an elementary proof , but any help would be appreciated!  Thank you. How do I insert a backslash inside a mathematical statement? I'm trying to write a set subtraction operation in an AMS formatted template.  In the end, what I want should look like A\B.  But I'm doing this in math-mode $A\B$ and this gives me errors.  I've also tried using the \diagdown symbol, but it doesn't work either.  Is there another way to do this? What is meant by a "random variable"? Use central limit to solve a problem I am asked to solve the following question using central limit theorem. In an election between two candidates, A and B, one million individuals cast theirvote. Among these, 2000 know candidate A from her election campaign and vote unanimously for her.The remaining 998000 voters are undecided and make their decision independently of each other byflipping a fair coin.Approximate the probability pA that candidate A wins up to 3 significant figures.  It's easy to solve directly. PA = $\frac{0.5(10000-2000)+2000}{10000}$ = 0.501. However, I am quite confused about how to solve this problem by central limit theorem. I'm trying to configure a ubuntu server network. [Edit] My ifconfig: lo: flags=73&lt;UP,LOOPBACK,RUNNING&gt; mtu 65536 inet 127.0.0.1 netmask 255.0.0.0 inet6 ::1 prefixlen 128 scopeid 0x10&lt;host&gt; loop txqueuelen 1000 (Loopback Local) RX packets 352 bytes 26160 (26.1KB) RX erors 0 dropped 0 overruns 0 frame 0 TX packets 352 bytes 26160 (26.1KB) TX errors 0 dropped 0 overruns 0 carrier 0 collisions 0  I edited the /etc/network/interfaces file and here is how it is: auto lo auto eth0 iface eth0 inet static address 192.168.15.141 netmask 255.255.255.0 gateway 192.168.15.1  I'm trying to restart the server and I'm using: sudo ifdown --exclude=lo -a &amp;&amp; sudo ifup --exclude=lo -a  This command is an answer from this link:  And I'm getting the following error: Cannot find device &quot;eth0&quot; Failed to bring up eth0.  I'm pretty lay in configuring a server, so I have no idea what's going on. Can someone help me please?