Prove by induction: $b^n-a^n=(b-a)\cdot \sum_{k=0}^{n-1}{a^k\cdot b^{n-1-k}}$ $\forall n\in \mathbb{N}$ and $a,b\in \mathbb{R}$
Induction Proof that $x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\ldots+xy^{n-2}+y^{n-1})$
$$f(x,y) = \left\{\begin{array}{cc} \frac{xy}{x^2+y^2} &amp; (x,y)\neq(0,0) \\ f(x,y) = 0 &amp; (x,y)=(0,0) \end{array}\right.$$  In order to verify if this function is differentiable, I tried to prove it by the theorem that says that if $\frac{∂f}{∂x}$ and $\frac{∂f}{∂y}$ exist and are continuous at the point $(x_0,y_0)$, then the function is differentiable at this point. So I did:  $$\frac{\partial f}{\partial x}(0,0) = \lim_{h\to 0}\frac{f(0+h,0)-f(0,0)}{h} = 0$$ $$\frac{\partial f}{\partial y}(0,0) = \lim_{h\to 0}\frac{f(0,0+h)-f(0,0)}{h} = 0$$  so we have that the partial derivatives at point $(0,0)$ is $0$. Now, if we take the derivative at $(x,y)\neq (0,0)$ and then take the limit of it as $(x,y)\to(0,0)$, we can see if the derivatives are continuous or not. So here it is:  $$\frac{\partial f}{\partial x}(x,y) = \frac{y(y^2-x^2)}{(x^2+y^2)}$$  but   $$\lim_{(x,y)\to(0,0)} \frac{y(y^2-x^2)}{(x^2+y^2)} $$ does not exist (by wolfram alpha... but can anybody tell me an easy way to prove this limit does not exist? easier than taking the limit in different directions?), therefore the derivative is not continuous at $(0,0)$, so we can't say $f$ is differentiable at $(0,0)$, but for $(x,y)\neq (0,0)$ the function is continuous, as it is a quotient of continuous functions. So $f$ is at least differentiable at $(x,y)\neq (0,0)$.   Now, to verify differentiability at $(0,0)$ I think we must use the limit definition of differentiablity: A function is differentiable at $(0,0)$ iff: $$\lim_{(h,k)\to (0,0)} \frac{f(0+h,0+k)-f(0,0)-\frac{\partial f}{\partial x}(0,0)-\frac{\partial f}{\partial y}(0,0)}{\|(h,k)\|} = 0$$  Let's calculate this limit:  $$\lim_{(h,k)\to (0,0)} \frac{f(0+h,0+k)-f(0,0)-\frac{\partial f}{\partial x}(0,0)-\frac{\partial f}{\partial y}(0,0)}{\|(h,k)\|} = \\ \lim_{(h,k)\to (0,0)} \frac{\frac{hk}{h^2+k^2}}{\sqrt{h^2+k^2}} = \\ \lim_{(h,k)\to (0,0)} \frac{hk}{(h^2+k^2)\sqrt{h^2+k^2}}$$  which I think, it's a limit that does not exist, therefore the function isn't differentiable at $(0,0)$ PhD in Europe without scholarship from non-European homeland I just did a clean installation of Ubuntu 18.04.1 on a tablet which was previously running Ubuntu 17.10. Back then just after the installation the screen was rotated, but I was able to get it to work (I think it was running on Wayland).  Now with 18.04.1 the situation is similar: when Ubuntu first boots the login screen is upside-down, but that's not a problem. This time, after login, the screen is correctly oriented but the mouse cursor is upside down and mouse movements are inverted with respect to both axes. The problem is that when I try to click on something the click is registered on the position on the other side of the screen (symmetrically with respect to the center).   It almost seems like the screen (desktop, application, etc.) is correctly displayed, with also the mouse positioned correctly and moving as it should, but then the mouse cursor which is shown to the user is inverted, like if the screen had been inverted just before drawing the cursor.  I already tried the answers I could find, but none worked for me. Anyone have an idea of what's going on?  Thanks Searching for Google contacts not in Groups Possible Duplicate:          In Google Contacts, I've had Groups set up which look like this: All Contacts.. Family.. School.. Work.. etc.  Each contact I have is assigned to All Contacts. In addition, it is assigned to at least one more Group.  Sometimes when I add contacts, I forgot to bucket them in the respective groups, so I was wondering is there anyway I could do a search for contact that belongs toAll Contact-group, but does not belong to any other group ? "Reflexive only" verbs English can use a lot of verbs in a reflexive context. Even ones that usually are used intransitively.     I laugh myself silly.   However, it seems like there are very few – perhaps no – verbs that ONLY work in a reflexive context.  Are there any? Suppose John has 5 sweets. Is there any difference between the following two sentences?     Jack has 3 times as many sweets as John.      Jack has 3 times more sweets than John.   I prefer the first construction and would know unambiguously that Jack has 15 sweets in this case. However in the second construction I would be inclined to think that Jack has 20 sweets, since it seems to suggest 15 sweets in addition to the original 5. Functions satisfying $f\left( f(x)^2+f(y) \right)=xf(x)+y$ Where does Minecraft for Windows 10 store its data? Overfitting a logistic regression model Proof by Induction: For any integer $n > 0$, $3|(5^{2n} − 1)$. Why does `find` not delete the files it found? I'm trying to understand why the find command is not deleting files in one particular case. I've got the following command:  find C:\\path\\to\\Pictures\\pmcctv -name cap_*.jpg -o -name cap_*.ogg -o -name cap_*.flv -o -name cap_*.mp4 -o -name cap_*.webm -mtime +7   Which returns the files I would expect:  C:\path\to/Pictures/pmcctv/cap_20160915T193251_620067800.jpg C:\path\to/Pictures/pmcctv/cap_20160915T193322_742708800.jpg C:\path\to/Pictures/pmcctv/cap_20160916T081046_394767500.jpg C:\path\to/Pictures/pmcctv/cap_20160916T081125_615129600.jpg C:\path\to/Pictures/pmcctv/cap_20160916T081129_503678200.jpg C:\path\to/Pictures/pmcctv/cap_20160916T081255_842394600.jpg C:\path\to/Pictures/pmcctv/cap_20160916T161008_693586800.jpg C:\path\to/Pictures/pmcctv/cap_20160916T161012_749396800.jpg C:\path\to/Pictures/pmcctv/cap_20160916T161043_774132200.jpg C:\path\to/Pictures/pmcctv/cap_20160916T161150_497251900.jpg C:\path\to/Pictures/pmcctv/cap_20160916T161154_437379600.jpg C:\path\to/Pictures/pmcctv/cap_20160916T161337_350955300.jpg C:\path\to/Pictures/pmcctv/cap_20160916T161341_452596000.jpg C:\path\to/Pictures/pmcctv/cap_20160916T161412_870774000.jpg C:\path\to/Pictures/pmcctv/cap_20160916T161443_969064400.jpg C:\path\to/Pictures/pmcctv/cap_20160916T161554_468109900.jpg C:\path\to/Pictures/pmcctv/cap_20160916T161558_378086000.jpg C:\path\to/Pictures/pmcctv/cap_20160916T161916_668433100.jpg   Now if I try to run the same command with -delete, the files are not deleted:  find C:\\path\\to\\Pictures\\pmcctv -name cap_*.jpg -o -name cap_*.ogg -o -name cap_*.flv -o -name cap_*.mp4 -o -name cap_*.webm -mtime +7 -delete   I've also tried with -exec rm, but they are also not deleted:  find C:\\path\\to\\Pictures\\pmcctv -name cap_*.jpg -o -name cap_*.ogg -o -name cap_*.flv -o -name cap_*.mp4 -o -name cap_*.webm -mtime +7 -exec rm {} \;   find returns no error and the files are owned by me with permissions 644.  Any idea what could be causing this?  (note: I've cut the commands to make it more readable but normally there's no line break) Is color undefined I want to check whether a color has been previously defined, consider the below code which doesn't execute.  \documentclass{article}  \usepackage{xcolor} \newcommand{\colorprovide}[2]{\ifx\undefined{#1}\colorlet{#1}{#2}\fi}  \begin{document}     %Initial Definition     \colorprovide{mycolor}{red}     \textcolor{mycolor}{Test 1}      %Will overwrite     \colorlet{mycolor}{green}     \textcolor{mycolor}{Test 2}      %Should NOT overwrite as it exists     \colorprovide{mycolor}{red}     \textcolor{mycolor}{Test 3} \end{document}   In other words and analogous to \providecommand{...} not renewing an existing command if it exists, I need a macro which will will NOT redefine a color if the color has already been defined.  The above code throws an error as the logical checking part \ifx\undefined... is incorrect. What is a simple way of downloading a file from a URL path? I have less than 3000 rep on StackOverflow. Now that the new closing changes are live, I cannot supply a custom off topic message:    I can enter a custom off topic flag message without issue here on meta, though it only has three canned off topic reasons, rather than five.  I'm not sure if this is the same for users with enough rep to vote to close directly, but the lack of a custom message eliminates the ability to use the correct flag reason in some cases.  For example, I wanted to flag , which is simply off topic on SO. However, I can't use a custom message, and none of the canned ones are appropriate.    appears to be the most up to date official information on custom reasons, and indicates that they are remaining in play for now.  supports that.  If this is by design, how should I flag such questions? This is not a case worth bothering the mods over, so I don't want to use a custom diamond attention flag. Word for lights on police cars, etc Onguard U-lock jammed I'm currently trying to unlock my bike that has been sitting on my porch for a while, the key works and turns, and I can hear the lock engaging/disengaging, however I cannot get the shackle to come out. I think their might be dust/sand stuck in the mechanism however I'm not sure how to get it off now to check. Close popup doesn't appear after editing the question without refreshing the page Seems like a bug:  The user:   Edits a question Then, he decide to close it. Clicks on the Close button. The popup doesn't appear. Refreshes the page. Click on the close button. The popup appears.   After editing in the link appears a hash: #. But I don't think that this is the problem because if I add the hash without editing the close popup appears.  It's minor, but I think that is good to be reported and solved.  Edit: The close popup doesn't open after we click an edit has been made to this post; click to load. Can I change some but not all Apple keyboard function keys (dim vs F1, bright vs F2)? I want to override F1 and F2 behavior, since I have external display and Mac can't control display's brightness. The rest of the keys should retain their non Fx  I know about "Use all F1-F12 keys as standard function keys.", but I don't want to change all of the keys to be F when pressed, just some of them. (For me single press music and volume control is necessary.)  Is there a way to pick some of the function keys to default to numbered function instead of the "Apple" definitions system wide? How can I revert a config file back to the originally installed version after I have edited it? I want to revert a file in /etc/ back to its originally-installed form. I have edited the file since it was installed. How can I revert this config file? Apt is smart enough not to overwrite edited config files, so how do I tell it that I want it to do so?  For argument's sake, let's say that I want to revert the file /etc/foo.conf from the package foo. how do you make commands not show up on the chat I am making a server, and I put a comparitor clock down to constantly have clear weather. How do I silence the commands?