If $f(x)$ is continuous at the point $x=0$ and for all real numbers $x$ and $y$, the function $f(x+y)= f(x) + f(y)$. Show that $f$ is continuous for all values of $x$.  Not sure where to begin this problem. Any info on it would be appreciated.
I am trying to prove that if I have $f:\mathbb{R} \to \mathbb{R}$ satisfying $\forall x,y\in\mathbb{R},f(x+y) = f(x) + f(y)$. Which is assumed continuous at $0$, that $f$ is continuous on $\mathbb{R}$  I am fairly sure it is, for that property seems to be a property of polynomials, and we know polynomials are continuous where defined(all reals)  Any ideas for rigor?  $|x-a| \lt \delta \implies |f(x+a)-f(x)-f(a)| \lt \epsilon$  No idea where to go from here, this is my first time doing $\epsilon-\delta$ stuff.
Can I put 700c tires on rims made for 27 x 1.25"? Strange noise when moving files When I move files, a small 'noise' noise comes through my headphones or speakers.  It only happens when moving files.  Is there a reason why moving files causes this slight static?  It only lasts for as long as the files are moving.  I'm not as much as annoyed as I am curious.  It is a Gigabyte board with a integrated sound card.  I'm running Windows 7.  I don't think it used to do this but I'm not quite sure. Would the One Ring have stayed hidden if Bilbo hadn't turned up? Why won't my computer go to sleep automatically? Windows 8 is set to sleep after 30 mins, and it used to work, but recently it's started refusing to sleep. (I can still manually ask it to go to sleep without any issue.)    I was having issues a while ago, but it was with my network adapter. That's since been disabled, so it's definitely not that:    I've checked to see what devices are able to wake up my machine, but it only appears to be my mouse:    Which is odd, because I haven't recently changed my mouse, and more confusing still: The monitor does go to sleep just fine. If it was actually the mouse keeping my system awake, I'm pretty sure the monitor wouldn't go to sleep.  I've checked my Wake Timers, and nothing:    I've also checked my existing requests...  UPDATE: I found something. What to do with it, I don't know...    Note: Even when /requests says that there's "NONE" under every category, my machine still won't sleep(!).    In short: How can I tell what's preventing my computer from Sleeping?    UPDATE: Ok, so I now have a few more pieces of the puzzle. I came back to my computer and it was ASLEEP! Lawks! It seems that the only times it doesn't sleep is if VLC Player is open, even if a video isn't actually playing.  UPDATE UPDATE: Ok, so it won't sleep sometimes when VLC Player ISN'T running, either. Bah! How to convert decimal to hexadecimal in JavaScript Checking if a double (or float) is NaN in C++ given value of matrix, getting it's coordinate If $q$ is coprime to $a$ then $a\mid nq-1,\,$ so $q$ is invertible mod $a$ I think the following is true:  If $q$ is coprime to $a$ then $a\mid (nq-1)$ for some $n\in\mathbb Z$.  I have started to first reducing $n\in\mathbb Z$ to $n\in\{1,\ldots,a-1\}$ and then considering all the cases $q = ka + j$ for $j=1,2,3,\ldots,a-1$. I stopped at $j=3$ and found that this is kinda messy. Does anyone have an elegant proof for this (maybe using some results from number theory)?  [Note: question has been generalized from prime $q\nmid a$ to $\,q$ coprime to $a$] I´m posting this because I have some active questions in SO and I´m leaving for the weekend and wont have internet until monday or tuesday.  SO is a very active site and I hate no to answer/thank/comment people who are kind enough to answer my questions.  I think an "away" indicator like an out of office reply would be a really nice feature to add for "politeness" sake. Let $M_n$ denote the vector space of all $n\times n$ real valued matrices and for any matrix $A\in M_n$ let $A^t$ denote its transpose. Define the linear map $T:M_n\rightarrow M_n$ as $$T(A)=A+A^t.$$ What is the minimal polynomial of $T$? What are the eigenvalues and the corresponding eigenspaces of $T$? What is the characteristic polynomial of $T$?  I learned about minimal polynomials of operators from $\textit{Linear Algebra Done Right}$ by Axler. In his description he explains how in order to find the minimal polynomial you must find the smallest positive integer $m$ such that $T^m$ is a linear combination of $I, T, T^2, \ldots, T^{m-1}$. That is, the smallest integer $m$ such that $$T^m = a_0I+a_1T+a_2T^2+ \cdots +a_{m-1}T^{m-1}.$$  As for the eigenvalues...since we are working with real-valued matrices, then $A$ and $A^t$ have the same eigenvalues, so if $\lambda \in \mathbb{R}$ is an eigenvalue of $A$ (and hence of $A^t$) wouldn't $2\lambda$ be an eigenvalue of $A+A^t$? I apologize if I am saying utter nonsense... What is wrong with defining \bal as \begin{align}? I'm trying to define two commands \bal and \eal instead of inserting \begin{align} and \end{align} in my document, but in the code below, I'll get some errors. What is wrong with it?    \documentclass{article} \usepackage{amsmath} \makeatletter  \newcommand{\bal}{%  \begingroup  \align% \@ifstar{\nonumber}{}%  }  \newcommand{\eal}{\endalign\endgroup} \makeatother \begin{document} \bal* x=y \eal ‎\end{document} I Roll To Seduce The Dragon! (or, Do doublings of proficiency bonus stack?) Lets say there exist some enterprising young bard 3/ sorcerer 1 named Kirk.   Deciding to uphold (long past) family tradition, he has chosen to go out and have a little half-dragon of his own.   He chose Persuasion as his expertise skill, and is of a dragon bloodline himself, gaining the Dragon Ancestry perk.  Do they stack? Not the same ability, and both say "Your proficiency is doubled" when they activate. Granted, it would only happen when dealing with dragons. But there are certainly worse times to ramp up your charm. Find the value of $\sum_0^n \binom{n}{k} (-1)^k \frac{1}{k+1}$ Find the value of $\sum_0^n \binom{n}{k} (-1)^k \frac{1}{k+1}$.  Writing out several terms, I think the answer is $1/(n+1)$, but I'm struggling to prove this. I would greatly appreciate it if someone could show how to prove this. How can I re-write the conditionals for this button I've been trying out React for a couple of days and I tried to build a button component with some conditionals. Everything works but I feel like I am not doing it the right way. Could anyone review my code and give me some advice and tell me what I am doing wrong or what i could do better. I think I am using too many return statements. I use Typescript for type checking. Would appreciate any help! import { Link } from &quot;gatsby&quot;;  interface Props {   content: string,   color: string,   path?: string,   type?: string }  export default function Button(props: Props): JSX.Element {   let elementType;    const checkProps = () =&gt; {     const availableColors = ['blue', 'grey', 'dark-grey', 'white'];     const borderColors = [''];     let buttonStyle;      // Check if props.color exists in availableColors     if(!availableColors.includes(props.color)) {       console.error(`&quot;${props.color} color key not available for button style`);       return;     }          buttonStyle = buttonStyles[`button--${props.color}`];      // Check if props.content is valid     if(!props.content) {       console.error(`&quot;${props.content} is not valid as content`);       return;     }      // Check if props.type exists     if(!props.type) {       elementType = &lt;button className={ buttonStyle }&gt;{ props.content }&lt;/button&gt;       return;     }      if(props.type !== 'link') {       console.error(`&quot;${props.type} is not a valid type, use 'link' instead or use nothing for default button`);       return;     } else {         elementType = &lt;Link to={ props.path } className={ buttonStyle }&gt;{ props.content }&lt;/Link&gt;     }   }    checkProps();    return (     &lt;React.Fragment&gt;       { elementType }     &lt;/React.Fragment&gt;   ); }``` Is there any evolutionary/physiological advantage to multi-lateral symmetry? Just wondering ; I get the impression a large number of multicellular organisms are multi-lateral in their physical structure. Why would evolution/mutation have retained multi-lateral symmetry in an organism? What advantage/s does symmetry provide an organism?  EDIT:  By multi-lateral symmetry I mean at-least bi-lateral symmetry (as in the case of many beings such as elephants, crabs and fishes), but perhaps more than bi-lateral too (starfish come to mind)  END EDIT: Is there a Google Drive client available? How can time dilation be symmetric? Execute a Java program from our Java program I used   Runtime.getRuntime().exec("_____")   but it throws a IOException as below:  java.io.IOException: CreateProcess: c:/ error=5   at java.lang.Win32Process.create(Native Method)   at java.lang.Win32Process.&lt;init&gt;(Win32Process.java:63)   at java.lang.Runtime.execInternal(Native Method   I don't know whether I have the problem with specifying the path or something else. Can anyone please help me with the code. REST Service API executor.executeAsync Call Ends in Failure Function in Sharepoint Hosted App Any Answers Will Help me to Forward and Gain Knowledge in SharePoint Development  While i am Trying to get the list Items Using Rest API Calls in SharePoint Hosted App Using Below Function  function loadingSamples()  {     var JobListName = "RRR Jobs #";     var executor;     // Initialize the RequestExecutor with the app web URL.     executor = new SP.RequestExecutor("/");      //Fetching the choice values from the Job Status Field, to show in the Dropdown field     executor.executeAsync     ({         url: "/_api/web/lists/getbytitle('" + encodeURIComponent(JobListName) + "')/fields/getbytitle('Job Status')?$select=Choices",         method: "GET",         headers: { "Accept": "application/json; odata=verbose" },         success: successHandlerStatus,         error: errorHandler     }); } function successHandlerStatus(data)  {     alert("sucesss") }  function errorHandler(data, errorCode, errorMessage)  {     alert("errorMessage") }   It's always going to error function named errorHandler With  Error Code as -1002 and Error Message as Bad Request and Inside Data -> Body , the Error message is  {     "error":{            "code":"-1,Microsoft.SharePoint.Client.InvalidClientQueryException",         "message":{             "lang":"en-US",             "value":"The expression \"web/lists/getbytitle('DWR Jobs \" is not valid."         }     } }   Note :    This is working code in Sharepoint Hosted App that previously hosted in another Windows server. Now i hosted this App in New windows server,it's successfully hosted but it gives above mentioned Error in Rest Service Call. Can a concentration spell be cast without actually concentrating on it for an "instant" effect? The book says:     Concentration. Some spells require you to maintain concentration in order to keep   their magic active. If you lose concentration, such a spell ends.   So what happens if, for example, you are already concentrating on a spell, and cast Witch Bolt without concentrating. Would you get the initial arc of blue energy, or would it not even go off?  I am asking if you can cast a spell that says concentration without concentrating for an immediate effect. This would be useful to avoid interrupting your existing concentration. In essence, if you are concentrating on thing A (doesn't have to be a spell since other things need concentration), can you cast a concentration spell getting an immediate effect without losing concentration on thing A?  Since the definition of concentration is to keep the spell active (meaning it is already active) I would think that you would get the initial effect (if it had one) without the need to concentrate on it.