I've stumbled upon that limit in the book that I'm studying (Spivak's Calculus). At the current stage of the book, this is a trivial task if $\alpha \in \mathbf Q$. But the problem statement doesn't specify anything about $\alpha$, except that $\alpha &gt; 1$, so what if $\alpha \in \mathbf R$? At the current stage there is no Chain Rule, L'Hopital's Rule, exp/log, integrals, series. I've attempted to show that $$\lim_{h^+ \to 0} \frac{|h|^\alpha}{h}=\lim_{h^+ \to 0} h^{\alpha - 1}=0=\lim_{h^- \to 0} -(-h)^{\alpha - 1}=\lim_{h^- \to 0} \frac{|h|^\alpha}{h}.$$ I've tried:  Something like $0&lt;h&lt;\delta = \epsilon^{\frac{1}{\alpha - 1}} \implies 0&lt;h^{\alpha - 1}&lt;\epsilon$, but I'm not sure how to prove the general statement that $\forall x,y,a \in \mathbf R, a &gt; 0, \ \ 0 \leq x &lt; y \implies x^a &lt; y^a$. Also I can only prove the existence of the roots like $\epsilon^{\frac{1}{n}}, n \in \mathbf N$. Prove that $f(x)=x^\alpha, \alpha &gt; 0$, is continuous. Using limits multiplication rule for that would not work for irrational $\alpha$. Showing continuity through differentiability is problematic, I can prove $(x^n)'=nx^{n-1}$ only for rational n (this also hinders the proof of the statement in 1., i.e. that $f(x)=x^\alpha, \alpha &gt; 0$ is an increasing function on $x \geq 0$).  Nevertheless, the problem is given at this stage, so what am I missing here?
I want to calculate the derivative of $|x|^\alpha$ with respect to $x$, where $1&lt;\alpha&lt;2$.  $\frac{d|x|^\alpha}{dx}=\alpha|x|^{\alpha-1}\mathrm{sign}(x)$  Is it correct?  And is it true that  $\lim_{x \to 0} \alpha|x|^\alpha\mathrm{sign}(x)=0$?  What with the following derivative $|x|^\alpha\rm{sign}(x)$. I would be grateful for any hints.
Bayes rule logic. Why we do use this? Removing files with strange names Possible Duplicate:          Somehow I ended up with a file named "-r".  How do I remove it?  rm -r doesn't work.  I tried 'rm -i `ls -a`' to step through the file names, but it didn't prompt me to delete this one.  Edit A very hacky approach was to use python's os.unlink function.  That worked, but I'm curious to hear other ways. tar compression without directory structure I know that I can use  to make cross-references and hyperlinks clickable.  That makes the clickable areas outlined in fluorescent green, however.  How can I make the green boxes go away? When is tension constant in a rope? Consider a massless rope with pulling forces applied at each end.  How do we decide if tension is constant or not in a rope? Consider a few example scenarios:  For example, if there is a knot in the rope the tension is not constant throughout (why?. Similarly, if the rope is hung over a cylindrical pulley of a non-negligible radius, the tension is not constant (why?). If there isn't anything touching the rope, for example, 2 people tugging on the rope at each end, the tension is constant (why?). How can I write the name show in axis of 3D array using TiKz I am drawing a 3D array like the attached picture, which actually describes a multidimensional array. However, I can not put the name of the axis and take the value from the matrix like the attached picture.  This is the code that I have:  \documentclass[margin=5mm]{standalone} \usepackage{tikz} \usetikzlibrary{matrix,calc}  \begin{document} \begin{tikzpicture}[every node/.style={anchor=north east,fill=white,minimum width=1.4cm,minimum height=7mm}] \matrix (mA) [draw,matrix of math nodes] { (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ };  \matrix (mB) [draw,matrix of math nodes] at ($(mA.south west)+(1.5,0.7)$) { (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ };  \matrix (mC) [draw,matrix of math nodes] at ($(mB.south west)+(1.5,0.7)$) { (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ };  \draw[dashed](mA.north east)--(mC.north east); \draw[dashed](mA.north west)--(mC.north west); \draw[dashed](mA.south east)--(mC.south east); \end{tikzpicture}   \end{document} How can I send mail from an iPhone application My system is not dual-boot, I run a standard Ubuntu desktop system "on the metal" (I think running it in a VM is the same).  Pressing c while booting does not cause the GRUB menu to appear (which I believe it does for a dual-boot system).  How can I get the GRUB-menu to present itself on a single-boot system? Background  Would like to center all tables within floats, without having to add \centering to each table.  Ideas  Tried the following:  \let\originaltabular\tabular \renewcommand\tabular{\centering\originaltabular}   It compiles, but the tables remain left-justified.  Question  How would you edit the Preamble of a LyX document to make all the floating tables be centred? Why can't I connect to Pokemon Go via mobile data? It connects to Pokemon Go via wi-fi perfectly fine, but when I try and connect to Pokemon Go via 3G it just gets stuck at the first bar on the loading screen with the eventual pop up saying "Failed to get player information from the server." My 3G connection works perfectly fine with every other app I have, and has been easily able to stream HD videos so I can't see connectivity with 3G being an issue.   It makes even less sense as once connected via wi-fi, the game easily continues to operate via mobile data once I leave the house. However once the app crashes (which it usually does after 15-30 minutes) I try to reload it on mobile data and it gets stuck on the loading screen. If I can find a local free wi-fi spot nearby I can re-connect easily, and then move back onto mobile data.  I've tried removing and reinstalling Pokemon Go and that didn't work. I also reset all my app preferences on my phone and that didn't work.  Not sure if this helps, my phone is a Huawei P8, Android version 5.0.1.  I signed up to Pokemon Go using my Google account. Gauss law and electric flux Can you please explain me what is electric flux. I know it is $\int\mathbf{E}\cdot\mathrm{d}\mathbf{S}$, but in many place its given flux means to flow. But how does electric field "flow"? Also if we assume its the electric field lines (can you also specify what exactly are these) are "flowing" how does $\int\mathbf{E}\cdot\mathrm{d}\mathbf{S}$ give the total electric field lines passing through that area? (can you please elaborate when you explain?)  Pls. I know it shouldn't come here but there is no page for electric flux. I installed Windows 7 fresh and installed SP1. Now, when I try to check manually for Windows Updates it just hangs on the Checking for updates screen.    I tried running the tools in , but this did not fix the issue either:  No matter what I do it just hangs on the "Checking for updates..." screen and goes no further. $\operatorname{Aut}(\mathbb Z_n)$ is isomorphic to $U_n$. I've tried, but I can't solve the question. Please help me prove that:  $\operatorname{Aut}(\mathbb Z_n)$ is isomorphic to  $U_n$. How to repair a broken .EXE file association After (hopefully) scrubbing viruses out of a Windows 7 installation (after deciding not to simply run over the laptop repeatedly with my car), I've got everything almost back to normal. The only lingering issue I have is that for my non-admin users, the ".exe" file extension doesn't work. That is, clicking on the various desktop application links results in a "How do you want to open this?" dialog.  I've been through the alleged registry fixing from "winhelponline" and that had absolutely no effect.  I've tried running "assoc" for the affected users, but it reports the .exe association to be "exefile" even though it persistently does not work.  Right-clicking on a desktop icon and then choosing "start" does successfully open an application, but that's clearly a terrible situation.  For my admin user, things seem to work fine.  What do I need to do to get things working? I have been developing a few python addins and I'm just wondering what is the best way to re-compile the addin anytime I edit the script.  Currently, if I make an edit to the script, it seems like I have to   Close ArcMap Go to My addin folder and double click the 'makeaddin' python file Double click the 'Python Addin File Restart ArcMap for the changes to come into effect   If I don't do these steps, then the edits aren't recognized. This gets pretty annoying and time consuming when I want to place maybe a simple print line somewhere or I do a small edit just to test a result.  Is there a better way of doing this? Using methods from another class in the main? How to Convert Persian Digits in variable to English Digits Using Culture? As a part of self study, I am trying to prove the following statement:  Suppose $X$ and $Y$ are topological spaces and $f: X \rightarrow Y$ is a map. Then $f$ is continuous if and only if $f(\overline{A})\subseteq \overline{f(A)}$, where $\overline{A}$ denotes the closure of an arbitrary set $A$.   Assuming $f$ is continuous, the result is almost immediate. Perhaps I am missing something obvious, but I have not been able to make progress on the other direction. Could anyone give me a hint which might illuminate the problem for me? Confusion about the definition of function Yesterday I was talking to one of my friends about the definition of function. The formal definition of function is given by Cartesian Products but my friend's question was whether it is possible to define a function without being acquainted with any concept of Cartesian Products.   To answer this question I told him the definition of a function may be given by defining a function $f$ from the set $X$ to $Y$, denoted by $f:X\to Y$, to be some "rule" (the word rule being an undefined concept) which associates, to each member of $X$, exactly one element of $Y$.     But my friend said that the definition using the concept of "rule" isn't rigorous and there are examples that don't obey the definition but still is a function in the Cartesian Product sense.    My questions are,   Why the definition of function using concepts such as "rule" isn't rigorous?  Is there really any example which doesn't obey the definition but still is a function in the Cartesian Product sense? Why doesn't the Enterprise use stealth/invisibility technology like the Romulans and Klingons? The stealth mode of starships plays a huge role in many Star Trek films and episodes and is a strong advantage in combat. I remember an episode revealing that the Federation developed a secret stealth technology (afair on Riker's former ship with another captain appearing in this episode) that even made it possible to fly with the ship through matter.   Nonetheless, while stealth technology seems standard for Romulans, Klingons (Ferengi, Vulcans?) and does not break any intergalactic laws (do these exist at all?), the Federation doesn't use it?