$$f(x,y)=\begin{Bmatrix} 0 &amp; (x,y)=(0,0)\\   \frac{xy}{x^{2}+y^{2}}&amp;(x,y)\neq (0,0)  \end{Bmatrix}$$   How to find out if this is continuous?  And do the partial derivatives exist at $(0,0)$?
I am aware there are many questions similar to this on MSE but I am having trouble following any of the solutions given.   I have the function given by   $f(x) = \dfrac{xy}{x^2+y^2}\;$ for $(x,y)\neq (0,0)$ and $f(0,0)=0$  I have calculated the partial derivatives and found:   $$\frac{\partial f}{\partial x} = \frac{y(-x^2+y^2)}{(x^2+y^2)^2}, \quad \frac{\partial f}{\partial y} = \frac{x(-y^2+x^2)}{(y^2+x^2)^2}$$  Now I need to show the partial derivatives exist for $(x,y) = (0,0)$.  I am also asked to show it is not continuous at $(x,y)=(0,0)$, based on other answers I've seen it seems like this follows from solving the first part but I fail to see how that follows through as well.   I could just copy the solutions with my function in place as I have seen a lot of answers using the definition of the derivative, but realistically I do want to understand what the thinking behind this is.   Any help would be appreciated.
How do I autohide the top panel in Unity? When windows are not maximized, the top panel is basically wasted space for me in Unity.  Is there a way to make it autohide? i.e. make it appear on mouse over. When I try to install any package through the command line, I get an error.   $ sudo apt-get install &lt;package&gt; Reading package lists... Done Building dependency tree        Reading state information... Done E: Unable to locate package &lt;package&gt;   Can anyone help me on this? $|f(x)-f(y)|\geq k|x-y|$.Then $f$ is bijective and its inverse is continuous. My exercise says: Let $f:\mathbb{R} \rightarrow \mathbb{R}$ a continuous function e suppose that exists $k$ such that:  $$|f(x)-f(y)|\geq k|x-y|$$  Then $f$ is bijective and  its inverse is continuous.  Well, there's a Theorem , Invariance of domain, that says   "If $U$ is an open subset of $\mathbb {R^n}$ and $f : U \rightarrow\mathbb{R}$ is an injective continuous map, then $V = f(U)$ is open and $f$ is a homeomorphism between $U$ and $V$".  But I'm not knowing how to proceed...need a clue...Thanks for attention!!! Is there a reason why the pin diagram of an RJ-45 looks like this?    I understand that the orange pair is used for Tx and the green pair for Rx. Why is it that the orange pair are kept together while the green pair are separated?  What are the consequences of messing around with this order as long as it is the same at both the ends? $|G|>2$ implies $G$ has non trivial automorphism Well, this is an exercise problem from Herstein which sounds difficult:   How does one prove that if $|G|&gt;2$, then $G$ has non-trivial automorphism?   The only thing I know which connects a group with its automorphism is the theorem, $$G/Z(G) \cong \mathcal{I}(G)$$ where $\mathcal{I}(G)$ denotes the Inner- Automorphism group of $G$. So for a group with $Z(G)=(e)$, we can conclude that it has a non-trivial automorphism, but what about groups with center? Can you use Uncanny Dodge against extra damage effects? Controlling bandwith while using GPRS on Windows 7 I have an issue with my windows 7 machine using a lot of bandwith while connected to the internet. I'm trying to limit the bandwith by using Opera in Turbo mode, but some other program is sucking down content and emptying my GPRS mobile plan.   Is there some way I can control which applications are allowed to use bandwith while on GPRS? Preferable only Opera should be able to use the network.   Thanks. On many occasions, I've found questions that should be moved to AskUbuntu, or Unix/Linux, but when I click the "flag" link, I don't have those as options. I hate to leave something flagged as off-topic with no suggestion of where to put it, so I just cancel.   What determines the sites that show up on that list? Is it best to flag them off topic even if the ideal site doesn't appear? Is there a quicker way to turn on WiFi Hotspot? is there a way to automate (or at least reduce the number of steps) to turn on or turn off the WiFi hotspot on my Galaxy 3.  I have to    unlock  press the menu button  press settings press More Settings Tethering and portable hotspot Slide portable hotspot on. Equation numbers are shown in reverse parentheses I am using   \begin{equation} a = b \end{equation}   to write the equations in my paper. However, the problem is that the number related to each equation is not shown in the correct format like the picture   How can I fix this?  This is what I have at the first of my code:  \usepackage{graphicx} \usepackage[justification=centering]{caption} \usepackage{float} \usepackage{url}% typeset urls \usepackage{times} \usepackage{url} \usepackage{latexsym} \usepackage{fontspec} \usepackage{multirow} \usepackage{listings} \usepackage{setspace} \usepackage{listings} \usepackage{fontspec} \usepackage{amsmath} \usepackage{polyglossia} \usepackage{color} \usepackage{bidi} \newfontfamily\faFont[Script=Arabic]{Yas}  \newenvironment{Fa}{\begin{RTL}\faFont}{\end{RTL}} \newcommand{\fa}[1]{{\faFont\RL{#1}}} \newcommand\Mark[1]{\textsuperscript#1} \definecolor{backcolour}{rgb}{0.95,0.95,0.92} \definecolor{darkblue}{rgb}{0.95,0.95,0.92} \setmainlanguage{english} \setotherlanguage{farsi} \newfontfamily\farsifont[Script=Arabic]{Yas} Soundness vs completeness, am I understanding? And proving soundness? I am getting FATAL_ERROR|System.QueryException: List has no rows for assignment to SObject. I know that it is because it is not returning any rows in the SOQL query but what I want is that if it returns any row, it should update ALO field on contact object, if it is null then ALO field should be set to null.  Here is my code:  ziptoLOD = [SELECT Id, Name, LOD__c             FROM Zip_3_to_LOD__c             WHERE Name =: contact.Extract_Zipcode__c ];  Zip_3_to_LOD__c ziptoLOD2 = (ziptoLOD != null) ? ziptoLOD : null;  if (ziptoLOD2 == null) {     contact.ALO__c = ' ';    } else {     System.debug(System.LoggingLevel.ERROR, 'Zip to LOD' + ziptoLOD2);     contact.ALO__c = ziptoLOD2.LOD__c; }   My code gets stuck at the SOQL query and gives me the list has no rows error. How to solve the equation $\phi(n) = k$? If I traceroute to my server via a bunch of different methods, the external ip seems to not work. However the server is online and accessible by this ip.  Localhost:  michaelxu@michaelxu-server:~$ traceroute 127.0.0.1 traceroute to 127.0.0.1 (127.0.0.1), 30 hops max, 60 byte packets  1  localhost.localdomain (127.0.0.1)  0.120 ms  0.069 ms  0.064 ms   Internal Ip:  michaelxu@michaelxu-server:~$ traceroute 192.168.1.3 traceroute to 192.168.1.3 (192.168.1.3), 30 hops max, 60 byte packets  1  michaelxu-server (192.168.1.3)  0.122 ms  0.065 ms  0.067 ms   External IP:   michaelxu@michaelxu-server:~$ traceroute 24.24.174.153 traceroute to 24.24.174.153 (24.24.174.153), 30 hops max, 60 byte packets  1  * * *  2  * * *  3  * * *  4  * * *  5  * * *  6  * * *  7  * * *  8  * * *  9  * * * 10  * * * 11  * *^C michaelxu@michaelxu-server:~$   ...it contines all the way to 30 ... the same thing happens if I traceroute it through the dns server  It also occurs if i do it from another online service:     Tracing route to cpe-24-24-174-153.socal.res.rr.com [24.24.174.153]...  hop rtt rtt rtt     ip address  fully qualified domain name 1   1   1   1       70.84.211.97    61.d3.5446.static.theplanet.com 2   0   0   0       70.87.254.5 po101.dsr02.dllstx5.networklayer.com 3   0   0   1       70.85.127.109   po52.dsr02.dllstx3.networklayer.com 4   0   0   0       173.192.18.230  ae17.bbr02.eq01.dal03.networklayer.com 5   0   0   0       66.109.9.97 ae-11-0.pr0.dfw10.tbone.rr.com 6   41  40  39      66.109.6.208    ae-6-0.cr0.dfw10.tbone.rr.com 7   39  39  40      66.109.6.0  ae-3-0.cr0.lax30.tbone.rr.com 8   37  39  39      66.109.6.65  9   42  40  39      72.129.9.3  agg1.lamrcadq-ccr02.socal.rr.com 10  36  36  35      72.129.10.73     11  53  60  49      24.24.209.18    ge17-0-spsdca1-ars1.socal.rr.com 12  *   *   *            13  *   *   *            14  *   *   *            15  *   *   *          What do the * * * mean? Is there any way to "fix this" How many combinations are possible in the game tic-tac-toe (Noughts and crosses)?  So for example a game which looked like: (with positions 1-9)  A1   --   B1  A2   --   B2  A3   --   --   [1][3][4][6][7] would be one combination How should ambiguous tags be dealt with? sudo apt update always giving Clearsigned file isn't valid, got 'NOSPLIT' (does the network require authentication?) The result of a Google search for &quot;error&quot; and &quot;PREP line&quot; is:  &quot;error&quot; &quot;on line&quot;: 439,000,000 results &quot;error&quot; &quot;in line&quot;: 60.600.000 results &quot;error&quot; &quot;at line&quot;: 35,800,000 results  While on is the indisputable number one, in and at have also very many hits. In the following context, which one should I use.  __ the first line you create an instance of XXX and XXX leads to the runtime error __ the last line.  I think, both prepositions are different. Although the word create refers to the action which is executed while runtime, the former part refers to the actual code that is written. In contrast, the latter part is figurative. I don't refer to the written code. The error does not really occur in the code but the code is the representation of the assembly which fails. Thus, I have two bests guess:  On the first line you create an instance of XXX and XXX leads to the runtime error at the last line. In the first line you create an instance of XXX and XXX leads to the runtime error on the last line. Passive with "to get" What is the function of the "Sticky edges" on/off setting in displays? I have installed Ubuntu 12.04 (Precise Pangolin). If I open System Settings &gt;&gt; Displays there is a new (?) on/off button for a setting named Sticky edges.  Can someone tell me what this setting is supposed to control?