Let $$f(x,y)=\left \{  \begin{array}{ll}   \frac{x^2+y^2}{sin(\sqrt{x^2+y^2})}  &amp; \mbox{if } 0&lt;||(x,y)||&lt; \pi \\   0 &amp; \mbox{if } x=(0,0)  \end{array} \right.$$  Determine if $f$ is differentiable at (0,0)  Im trying to use that if the partial derivatives exists and are continuous at (0,0) then f is differentiable at (0,0) but i can't conclude. I don't even know if I'm in the right way
Take the function  $$ \begin{cases}f(x,y)=(x^2+y^2)/\sin(\sqrt{x^2+y^2}) &amp; \text{when } 0&lt;\lvert (x,y)\rvert&lt;\pi\\   0 &amp; \text{when } (x,y)=(0,0) \end{cases}$$  I got that this function is differentiable at $(0,0)$. I calculated the partial derivatives at $(0,0)$ of which both were $0$ and trying to show the definition of differentiability is not satisfied but I keep getting that it is. Plz help.
Suppose I define a new theorem "Theorem" and declare that its counter should be reset whenever the subsection counter is incremented or reset:  \newtheorem{theorem}{Theorem}[subsection]   At some point in my (article) document I start a new \section (say, section 2) without starting a \subsection. I expect the first theorem in this section to be numbered with "2.0.1" (which may be considered bad style). However, the theorem is numbered "2.0.2".  Is this "by design" (technically, I didn't start a new \subsection) or a bug? Moreover, what would be a proper workaround? (Such a workaround should not involve manually resetting the theorem counter, and it should work for whatever theorem type I may define in addition to "Theorem".)  \documentclass{article}  \newtheorem{theorem}{Theorem}[subsection]  \begin{document}  \section{bla}  \subsection{blubb}  \begin{theorem} Some text. \end{theorem}  \subsection{foo}  \begin{theorem} Some text. \end{theorem}  \section{bar}  \begin{theorem} Some text. \end{theorem}  \end{document}   EDIT: In response to Seamus' comment: The unexpected counter value also occurs with amsthm and ntheorem. Highlight elements in the matrix I have a matrix and need to highlight some elements in the matrix. I have found how to highlight a single element (I want to put the cell in the square):  $$ A = \begin{bmatrix} \fbox{0} &amp; \fbox{0} &amp; 0 \\  \fbox{0} &amp; \fbox{0} &amp; 0 \\ 1 &amp; 1 &amp; 1 \\  \end{bmatrix}$$   But how can I outline elements of the submatrix like in the picture below? I need just a frame where some elements are located. SELECT GETDATE()   Returns: 2008-09-22 15:24:13.790  I want that date part without the time part: 2008-09-22 00:00:00.000  How can I get that? I think the title says it all. Let's say you're mastering a game without the benefit of miniatures, maps, or any kind of physical representation of the environment. How would you keep track of details like line-of-sight, ranged attack viability, and all the other small nuances which go with creating a "believable-enough" environment in which your PCs live and thrive?  (I admit that @LoganMacRae, who is my own DM, is an inspiration for this question. He does this with panache, and in my one-off, I feel like I have some mighty big shoes to fill!) I'm positive that vs I strongly believe that In general, which one is conveys a stronger conviction/opinion? I did some quick google search but didn't find any interesting results.      I'm positive that   or     I strongly believe that I installed CCSM to enable window previews in Unity. But when I went into extras, there was no 'window preview' option.  So I tried experimenting with other options in Compiz and now there are no borders in the dialogue boxes (close, minimize, restore buttons are not there, I had enabled some 'flat....' option in Compiz.  Later, I uninstalled Compiz and then reinstalled it. Now it says      "Failed to load session Ubuntu"   I can't even access through the guest account, I am new to Ubuntu.(please explain in detail).  All I have access to is the login screen (it still shows my custom wallpaper). Please help, I don't want to lose my stuff. I have had several business loans and understand that this is not by any means typical.  I noticed that for a year my payments were not decreasing the principal on the loan. When I asked the bank why, I found out that at some point my payments were getting applied to principal only. Now I am paying interest off that was not paid for 6 months or so.  My banker states that he can change all the interest payments to principal payments. I am confused to why this would be. If I make all my principal payments first then would the bank lose a substantial amount of interest?  This has happened on 2 of the 3 loans I have looked at so far.  If they are willing to change all the payments to principal only, would this be beneficial to paying less interest ultimately? I assume there would be an interest account accruing somewhere that would need to be paid off at the end of the loan.  I understand that typically you have a loan balance that would include interest and principal. My balance owed did not change when I continued to make same payment? It gets applied to interest. The balance on the statement must not be a true balance. There must be interest owed above the balance showed but why would they do this to start with? They applied full payments to principle only first in order for interest to get to be more than several payments A common scenario we run into on Stack Overflow is that we have a troll, spammer, or obvious sock puppet of another user and moderators decide to delete this user. We go on about our business only to find out an hour or two later that the account we just deleted has been recreated, using the same credentials, and is now back to trolling, spamming, or committing voting fraud once again.  I've seen some people recreate an account like this using the same credentials four or five times (multiple times in the last week, in fact). The only thing that stops them is if we remember to suspend the account for right before we destroy it. Accounts recreated using the same credentials as an account that was suspended will be locked into the same suspension the old account was serving.  In particular, this becomes a real problem with sock puppet accounts that have a network association reputation bonus attached to them, because the recreated accounts keep getting that 100 point bonus and can immediately be used for voting fraud, to troll chat, or to spam comments. I've had to deal with quite a few puppet accounts like this by suspending them for a year before deletion.  I propose that all moderator-deleted accounts behave as if they were suspended for a year upon deletion, so that recreating these accounts using the same credentials will cause the new account to be locked in a one-year suspension. This would prevent the new accounts from being used to abuse the site in the same manner as their old account did.  This won't put an end to people creating new spam, troll, or puppet accounts after we delete them, but it will at least slow them down by forcing them to create new login credentials each time. It also would curb abuse of account association bonuses by puppets. How to create a cross-browser sub document with iframe or shadow dom? How to set font to Arial throughout the entire document? In preamble, I have:  \renewcommand{\familydefault}{\sfdefault} \usepackage{arial}   but it doesn't work.  How do I set my font to Arial throughout the entire document?  UPDATE:  In short, both answers work.  It really depends on the environment. If using LaTeX and don't mind using a cloned font then choose Herbert's answer. If using XeLaTex and want to use the authentic font then choose Leo's. Differentiating $y=x^{2}$ I am reading in a book about differentiating, but I am confused with one of the steps he takes.  We start with:  $$ \begin{align} y &amp;= x^{2} \\ y + \mathrm{d}y &amp;= (x + \mathrm{d}x)^2 \\ y + \mathrm{d}y &amp;= x^2 + x\mathrm{d}x + x\mathrm{d}x + (\mathrm{d}x^2) \end{align} $$  Now the author simplifies this to:  $$y + dy = x^2 + 2x\mathrm{d}x + (\mathrm{d}x^2)$$  I dislike how the middle term is simplified to $2x\mathrm{d}x$ instead of $2(x\mathrm{d}x)$, as I feel like it is more intuitive on what is going.  As in, $2$ of the term $x\mathrm{d}x$, instead of $2x\mathrm{d}x$.  But I fear writing it as $2(x\mathrm{d}x)$ may result in an incorrect distributive property.  Next, he omits the $(dx^2)$:  $$y + \mathrm{d}y = x^2 + 2x \mathrm{d}x$$  Subtract the original $y = x^2$:  $$\mathrm{d}y = 2x \mathrm{d}x$$  Now here is where I get confused:  $$\frac{\mathrm{d}y}{\mathrm{d}x} = 2x$$  How can he just divide both sides by $\mathrm{d}x$!?  If the original term was $2$ of $x\mathrm{d}x$, wouldn't it have to be written out as $2x * 2\mathrm{d}x$, and thus divide both sides by $2\mathrm{d}x$ instead?  I think the root of my confusion is how to properly simplify: $$x\mathrm{d}x + x\mathrm{d}x$$  I trust that he is right, but I am looking for an explanation of why his simplification can work, and why $2(x\mathrm{d}x)$ would be incorrect!  Thank you! What options do I have to replace a failed service panel? What are my options to replace an old service panel?   We had a terminal get burned up in our old 1960s GE (TRX2012S if it matters, it's a split-bus/rule-of-six panel).   The tech who looked at it said the part is no longer available, but that if I could find another old service panel just like it, he could swap in the part. Otherwise, it's going to be $2,000+ to replace it with a modern service panel, which also requires getting up to modern code, which involves a lot of other stuff.   EBay and Google isn't showing any direct replacements (or the flush-mount TRX2012F).    What other options would there be to replace the panel internals?  Are the screw mounting holes compatible with modern GE panel internals? Is the main breaker separable and could that be changed alone?    Enclosure dimensions are about 20.5" x 12". Front panel dimensions are about 22.25" x 14".  UPDATE:    I'm looking at information on replacing the breaker box myself. As the homeowner I wouldn't be required to update anything else to get up to code. Does this sound like a good idea? I've done minor electrical DYI work but nothing this involved.  PICTURES:            More pictures: Can open sets and closed sets of $\mathbb{R}^n$ be homeomorphic? Suppose $A$ is an open subset of $\mathbb{R}^n$ considered as a topological subspace. Suppose $B$ is a another susbet of $\mathbb{R}^n$, and under the induced topology, is homeomorphic to $A$.  If $B$ is closed and bounded, then by the Heine-Borel theorem, $B$ is compact as a topological space so $A$ and $B$ cannot be homeomorhpic, because $A$ is not compact. But if $B$ is not bounded, what can be said about $B$, if any? Must $B$ be open, can it ever be closed (let's leave out the only case of being both open and closed--the entire ambient space $\mathbb{R}^n$), or can it be neither open nor closed? How does the kernel prevent a malicious program from reading all of physical RAM? If I write a program that tries to read memory at every possible address, and I run it on a "full" Unix, it will not be able to access all of the physical RAM.  But how does the operating system prevent it from doing so?  I am more familiar with small CPU architectures where any piece of assembly code can access everything. I don't understand how a program (the kernel) can detect such malicious operations. TRON: Game A.I algorithm? What is the difference between "citizen" and "denizen" Citizen: 1. A legally recognized subject ornational of a state or commonwealth, either native or naturalized. 2. An inhabitant of a particular town or city.  Denizen: 1. An inhabitant or occupant of a particular place.  Same thing? Profile changes in Q&A are not pushed to Area 51 using gb4e keeping interlinear glosses together on the same page I am using gb4e for several hundred interlinear glosses in my dissertation. Several of the examples span more than one page, that is, the vernacular and gloss will be on one page, while the free translation appears at the top of the next page.   How can I keep all three elements together on the same page? I often have subexamples (a, b, c) that don't need to stay together. That is, 1a can be on one page, and 1b on the next, as long as vernacular-gloss-free translation for each subpart stay together.  In the MWE below, the free translation is at the top of page 2, while the vernacular and gloss are at the bottom of page 1.  \documentclass[11pt,letterpaper]{article} \usepackage{lipsum} \usepackage{gb4e} \noautomath \begin{document} \lipsum[1-4] \begin{exe}     \ex \gll wuz x̃ũch it-ak xohish-em\\              1\textsc{sg.prn} bread eat-\textsc{inf} want.\textsc{prs}-1\textsc{sg}\\         \glt `I want to eat bread.'  \end{exe} \end{document}   A solution was proposed to a similar question (), but it requires me to manually look for what could be dozens of orphans throughout my dissertation (and doesn't work when applied to my example, since it still strands the FT). I would prefer a more elegant solution that takes care of the problem automatically. Can I enter the Schengen area with a German long-term visa through a different country? I am an Indian national. Right now I am staying in Israel. I will be moving to Germany on the last week of this month with german national visa directly from tel aviv. Ticket price of direct flight is high. So I am thinking to buy either Air France or LOT for travelling having layover at Paris or Poland. Am I allowed to do that? I mean is it mandatory to enter in Germany first? May I travel with these airlines with proper transit visa? We walk or ride on our vehicles to our destinations daily. Does our movement have any effect on the rotation of the earth according to Newton's law? What will be the effect if we move all the peoples along with their vehicles at their maximum velocity in one line in one direction along the equator? How much effect will it make?