How one can use a `\newcommand` for some formulas?
Faster way of writing \mathbf
Proving that $\lim_{x \to 0} \frac{f(x)}{|x|}=1$ implies $f$ is not differentiable at $0$. $$\lim_{x \to 0} \frac{f(x)}{|x|}=1$$ Then prove that $f$ is not differentiable at $0$. I was given this assignment and I'm not sure about my proof.. it feels like I'm assuming too much and not explaining with enough detail. Is there anything I need to add to make it sufficient? Assume toward contradiction that $f$ is differentiable at $0$. Then by definition we know that $f$ is continuous at $0$. Therefore $\lim _{x\to 0}f\left(x\right)=f\left(0\right)$ We know that $\lim _{x\to 0}\frac{\left(f\left(x\right)\right)}{\left|x\right|}=1$ Therefore we can conclude $\lim _{x\to 0}\frac{\left(f\left(x\right)\right)}{\left|x\right|}\cdot \left|x\right|=1\cdot 0=0$ But we can also rewrite this as $\lim _{x\to 0}\left(f\left(x\right)\right)\cdot \frac{\left|x\right|}{\left|x\right|}=f\left(0\right)\cdot 1=0$ Therefore $f(0) = 0$. Since $f$ is differentiable at $0$ we have $\lim _{x\to 0}\frac{\left(f\left(x\right)-f\left(0\right)\right)}{x-0}=\lim _{x\to 0}\frac{\left(f\left(x\right)-0\right)}{x}=\lim \:_{x\to \:0}\frac{\left(f\left(x\right)\right)}{x}\cdot \frac{\left|x\right|}{\left|x\right|}=\lim \:\:_{x\to 0}\frac{\left(f\left(x\right)\right)}{\left|x\right|}\cdot \:\frac{\left|x\right|}{x}$ Since the one sided limits of $\frac{f\left(x\right)}{\left|x\right|}$ exist, and are both equal to 1, we have $\lim \:_{x\to 0^+}\frac{\left(f\left(x\right)\right)}{\left|x\right|}\cdot \:\:\frac{\left|x\right|}{x}=\lim \:_{x\to \:0^+}\frac{\left(f\left(x\right)\right)}{x}\cdot \:\:\:\frac{x}{x}=1\cdot 1=1$ $\lim \:_{x\to 0^-}\frac{\left(f\left(x\right)\right)}{\left|x\right|}\cdot \:\:\frac{\left|x\right|}{x}=\lim \:_{x\to \:0^-}\frac{\left(f\left(x\right)\right)}{-x}\cdot \:\:\:\frac{-x}{x}=1\cdot -1=-1$ Since the limit doesn't exist, we have a contradiction to the fact $f$ is differentiable at $0$. Therefore $f$ is not differentiable at 0. I am setting up a server where there are multiple developers working on multiple applications.  I have figured out how to give certain developers shared access to the necessary application directories using the setgid bit and default ACLs to give anyone in a group access.  Many of these applications run under a terminal while in development for easy access. When I work alone, I set up a user for an application and run screen as that user. This has the downside that every developer to use the screen session needs to know the password and it is harder to keep user and application accounts separate.  One way that could work is using screen multiuser features. They do not work out-of-the-box however, screen complains about needing suid root. Does giving that have any downsides? I am pretty careful about using suid root anything. Maybe there is a reason why it isn't the default?  Should I do it with screen or is there some other intelligent way of doing what I want? Why, oh why, don't the Weasleys "conjure" up some diamonds/gold/etc, sell 'em to Muggles, and get rich? They could buy a mansion (like the Malfoys), and live there happily ever after.  I know there's  that prohibits one from conjuring food (which doesn't make much sense to me, either). Even if it applied to gold, what about all the other elements/compounds/materials? Could the law really be 'don't let a wizard conjure something if it'll make them richer'? That's totally nonsensical.  Why does Wizard society still have poor people, especially if they're talented? In $Sets$, the initial object is $0 = \emptyset$. We have $C \times \emptyset = \emptyset$ for any set $C$. As there are no maps to the empty set, we don't get much of the universal property of the product.  I am now wondering what is special about arbitrary categories $\mathscr{C}$ with $C \times 0 \cong 0$ for all objects $C$ of $\mathscr{C}$, where $0$ is the initial object of $\mathscr{C}$.   From Sheaves in Geometry and Logic by MacLane (p.194), in any topos $\mathscr{C}$, any arrow $k:C \to 0$ is an isomorphism. The universal property of the product would then also not yield very much.   Are there any categories of higher interest? Let's assume a very simplified situation. There is only one seller asking for 10$ and there is only one buyer bidding 20$. At what price would the transaction go?   Some answers say it won't, because the ask price has to match the bid price. () Other answers say, that the time ordering is important. () Which answer is correct? What is the final price in the presented situation? How many actuations are "too many actuations"? When looking at used camera bodies, a question that comes up a lot, is "how many shutter actuations, or clicks, does it have?"   What I am wondering is, how many is too many. In other words, is there a reference for knowing when a sensor is more likely to go? How should I judge a camera body with 20000 vs 5000? Is that a significant difference, or is that not even relevant.  Obviously this might something that is dependent on model, so if there is some information based on model that would be even more interesting.  One of the guys at Canon Tech Support said that this is completely irrelevant for the Rebel series cameras, but he is obviously slightly biased. :) Number of finite simple groups of given order is at most $2$ - is a classification-free proof possible? Entering Unicode characters in LaTeX How do I enter Unicode characters in LaTeX?  What packages do I need to install and what escape sequence do I type to specify Unicode characters in an ASCII source file? How do I make the method return type generic? Consider this example (typical in OOP books):    I have an Animal class, where each Animal can have many friends. And subclasses like Dog, Duck, Mouse etc which add specific behavior like bark(), quack() etc.  Here's the Animal class:  public class Animal {     private Map&lt;String,Animal&gt; friends = new HashMap&lt;&gt;();      public void addFriend(String name, Animal animal){         friends.put(name,animal);     }      public Animal callFriend(String name){         return friends.get(name);     } }   And here's some code snippet with lots of typecasting:  Mouse jerry = new Mouse(); jerry.addFriend("spike", new Dog()); jerry.addFriend("quacker", new Duck());  ((Dog) jerry.callFriend("spike")).bark(); ((Duck) jerry.callFriend("quacker")).quack();   Is there any way I can use generics for the return type to get rid of the typecasting, so that I can say      jerry.callFriend("spike").bark(); jerry.callFriend("quacker").quack();   Here's some initial code with return type conveyed to the method as a parameter that's never used.   public&lt;T extends Animal&gt; T callFriend(String name, T unusedTypeObj){     return (T)friends.get(name);         }   Is there a way to figure out the return type at runtime without the extra parameter using instanceof? Or at least by passing a class of the type instead of a dummy instance. I understand generics are for compile time type-checking, but is there a workaround for this? Can I change my Ubuntu Desktop into a different flavour (like Kubuntu)? Can I convert Ubuntu into another form of Ubuntu like Kubuntu or Ubuntu Studio and the system recognize itself as the new form?  UPDATE: After getting an answer and talking with people, I managed to gain a lot of information on this topic. I then wrote an article about converting between Ubuntu systems - How to pass parameters to an alias? For bash script, I can use "$@" to access arguments. What's the equivalent when I use an alias? Set the size of pgf picture? I have a pgf picture which I insert with:  \input{m1.pgf}   But the image is too big. Instead of regenerating a smaller image, I'd rather take advantage of vector graphics, and have XeLaTeX scale it. How do I set the width of pgf picture (keeping its aspect ratio)?  MWE  tex:  \documentclass{article}  \usepackage{fontspec} % enagles loading of OpenType fonts \usepackage{polyglossia} % support for languages  % fonts: \defaultfontfeatures{Scale=MatchLowercase,Mapping=tex-text} % without this XeLaTeX won't turn "--" into dashes \setmainfont{DejaVu Sans} \setsansfont{DejaVu Sans} \setmonofont{DejaVu Sans Mono}  % pgf: \usepackage{pgf}  \begin{document}  \input{m1.pgf}  \end{document}   Sample pgf:  (it's too big to cite it here, and I can't make much smaller). How to get directory size in PHP Some issues with timezones in PHP have been in the back of my mind for a while now, and I was wondering if there are better ways to handle it than what I'm currently doing.  All of the issues revolve around reformating database stored dates:  When dealing with a site that has to support multiple timezones (for users), to normalize the timezone offest of stored timestamps I always store it with the server timezone using the CURRENT_TIMESTAMP attribute or the NOW() function.   This way I don't have to consider what timezone was set for PHP when the timestamp was entered (since PHP time functions are timezone aware). For each user, according to his preference I set the timezone somewhere in my bootstrap file using:  date_default_timezone_set($timezone);   When I'm looking to format dates with the php date() function, some form of conversion has to take place since MySQL currently stores timestamp in the format Y-m-d H:i:s. With no regard to timezone, you could simply run:  $date = date($format,strtotime($dbTimestamp));   The problem with this is that date() and strtotime() are both timezone aware functions, meaning that if the PHP timezone is set differently from the server timezone, the timezone offset will apply twice (instead of once as we would like).  To deal with this, I usually retrieve MySQL timestamps using the UNIX_TIMESTAMP() function which is not timezone aware, allowing my to apply date() directly on it - thereby applying the timezone offset only once.  I don't really like this 'hack' as I can no longer retrieve those columns as I normally would, or use * to fetch all columns (sometimes it simplifies queries greatly). Also, sometimes it's simply not an option to use UNIX_TIMESTAMP() (especially when using with open-source packages without much abstraction for query composition).  Another issue is when storing the timestamp, when usage of CURRENT_TIMESTAMP or NOW() is not an option - storing a PHP generated timestamp will store it with the timezone offset which I would like to avoid.  I'm probably missing something really basic here, but so far I haven't been able to come up with a generic solution to handle those issues so I'm forced to treat them case-by-case. Your thoughts are very welcome What affects field upgrades in BF4? Any thoughts on a semidecadian badge for people with five years service and an average of N points per year? Given that Stack Overflow is close to its fifth birthday as a public site, has any thought been given to a 'semidecadian' badge for people who have:   been registered with the site five years (a semi-decade, hence the name), scored more than N points per year on average (where N might be 1000).   The idea behind 'average of N points per year' is to allow slow starters to gain the badge eventually, rather than ruling them out in year 1.  The details are obviously subject to tweaking.  This might be a gold badge (the platinum version would be the decadian badge, with a larger value of N required — and no-one eligible for another 5 years or so).  These would complement the current Yearling badge (which is silver). What is the difference between $* and $@? A particle experiences an acceleration described by  $$ a=kx^{-2} $$ where x is the displacement from the origin and k is an arbitrary constant.  To what value does the velocity v of the particle converge to as x approaches infinity if the particle starts at some point x0?  If I approach this problem with energy, then $$ W = \int F \mathrm{d}x $$ $$  = \int_{x_0}^\infty mkx^{-2} \mathrm{d}x $$ $$  = mk(-\infty^{-1}+{x_0}^{-1}) $$ $$ W = K = mk{x_0}^{-1} $$ $$ \frac{1}{2}mv^2 = mk{x_0}^{-1} $$ $$ v = \sqrt{2k{x_0}^{-1}} $$  How would I solve this problem with pure kinematics? (there appears to be some sort of cyclical dependency where acceleration affects velocity, velocity affects displacement, and displacement affects acceleration)  Likewise, two particles experience accelerations described by $$ a_1=k_1x^{-2} $$ $$and $$ $$ a_2=-k_2x^{-2} $$ where x is the distance between the two particles  What two velocities do the particles reach as x approaches infinity if the two particles are initially separated by some x0? Inserting data to lookup field using rest API In alert I am getting value in data.d.results[0](This is the value that I am getting from other list using rest).Subsystem.Title and also if I use only title column then the value is inserted in the list but when I use a lookup column i.e. Subsystem, then it goes to error function.  $.ajax({         url: "https://sharepoint.com/teams/_api/web/lists/GetByTitle('Features')/Items",         type: "POST",         data:  JSON.stringify({ '__metadata': { 'type': 'SP.Data.FeaturesListItem' },    'Title': data.d.results[0].Title,'Subsystem':data.d.results[0].Subsystem.Title}),         headers: {              "accept": "application/json;odata=verbose",             "content-type":"application/json;odata=verbose",             "X-RequestDigest": $("#__REQUESTDIGEST").val()         },         success: function(d) {alert("Items Inserted!");          //window.location.href=referrer;             },         error: function() {alert("Failed to Insert Items!");} }); pygame installation issue in mac os