Problem Let $f:\mathbb{R}\to\mathbb{R}$ such that $f$ is continuous and injective. Prove (or disprove) that $f$ is a strictly monotone function.  I have tried to prove the monotonocity of $f$ for then by the injectivity of $f$ strict monotonicity will follow. However, I couldn't make any success. I also tried to show that,  Let $f:\mathbb{R}\to\mathbb{R}$ such that $f$ is not a strictly monotone function. Then show that $f$ is either discontinuous or not injective.  In this case also I couldn't proceed beyond some trivial observations. Can anyone help me to prove it?
Show that if $f: \mathbb{R} \to \mathbb{R}$ is a continuous injective map, then it is strictly monotonic.    Could someone give me a proof for this? I have the intuition for why it's true - I'm just having trouble expressing that intuition in a rigorous manner. Basically consider two points $x_1, x_2 \in \mathbb{R}$. By the problem statement, $f$ is continuous on $[x_1, \, x_2]$. WLOG, assume that $f$ is strictly increasing. It there exists a point where it is not increasing, then $f$ hits a value twice, and it's not injective.
How to link to apps on the app store I am creating a free version of my iPhone game. I want to have a button inside the free version that takes people to the paid version in the app store. If I use a standard link    the iPhone opens Safari first, and then the app store. I have used other apps that open the app store directly, so I know it is possible.  Any ideas? What is the URL Scheme for the app store? Where are the developer options in Android 4.2? Completing a 3x3 correlation matrix: two coefficients of the three given I was asked this question in an interview.  Lets say we have a correlation matrix of the form \begin{bmatrix}1&amp;0.6&amp;0.8\\0.6&amp;1&amp;\gamma\\0.8&amp;\gamma&amp;1\end{bmatrix}  I was asked to find the value of gamma, given this correlation matrix. I thought I could do something with the eigenvalues, since they should be all greater than or equal to 0.(Matrix should be positive semidefinite) - but I don't think this approach will yield the answer. I am missing a trick.  Could you please provide a hint to solve for the same? Finite group for which $|\{x:x^m=e\}|\leq m$ for all $m$ is cyclic. Let $G$ be a finite group. For each positive integer $m$, if $x^{m}=e$ has at most $m$ solutions in $G$, $G$ is cyclic.   What I have thought is that $n=\sum_{d\mid n}\phi(d)$ can be used to solve this and showing that $|G|$ order element in $G$ exists is enough. I have quite a bad habit of copy-and-pasting my packages from one LaTeX document to another. Is there any way we can detect unused packages (i.e. \usepackage commands) so that they can be removed from the LaTeX document? TOC entries and labels for included PDF pages I am embedding full pages from another PDF document (using the pdfpages package), but I would like to be able to declare my own section headings before each embedded document (for e.g. hyperlinking / TOC purposes). pdfpages has an experimental feature which lets me do this, but the problem is that I want to put all of these embedded documents inside a section, so I need to insert a section heading and a subsection heading on the first document I embed. Unfortunately pdfpages doesn't seem to be able to do this. Is there a way for me to do this?  Edit: Thank you for the suggestions. I am also in need of something similar for general labels — at the moment I am abusing the addtolist option of \includePDF but the objects I'm labelling aren't really figures or tables. Rules and guidelines for drawing good schematics There are a lot of poorly drawn schematics here. A few times people have actually asked for critiques of their schematics. This question is intended as a single repository on schematic drawing rules and guidelines can point people to. The question is  What are the rules and guidelines for drawing good schematics?     Note: This is about schematics themselves, not about the circuits they represent. How can I test whether a command's output is an empty string while not   interfering with buffering?  Example:  command | ??? if [ $? = 0 ] &amp;&amp; [ -n "$output_from_command"]; then    ...   I want the output from the code above to be the same as when running command. How do I change permissions on a FAT32 formatted drive? I would like to update VirtualBox on my 32 bit Ubuntu 15.04 machine from 4.3.26 to 4.3.28, using a Debian package I manually downloaded from . But simply double-clicking the package opens up the Software Centre which gives an error that installing it would break an existing package (the exact message was:   Breaks existing package 'virtualbox' conflict: virtualbox-4.3 ()   and it does not appear to have an option to ignore this and proceed with the installation anyway. Is there a command-line work-around this problem, perhaps? I know there are many ways to prevent image caching (such as via META tags), as well as a few nice tricks to ensure that the current version of an image is shown with every page load (such as image.jpg?x=timestamp), but is there any way to actually clear or replace an image in the browsers cache so that neither of the methods above are necessary?  As an example, lets say there are 100 images on a page and that these images are named "01.jpg", "02.jpg", "03.jpg", etc. If image "42.jpg" is replaced, is there any way to replace it in the cache so that "42.jpg" will automatically display the new image on successive page loads? I can't use the META tag method, because I need everuthing that ISN"T replaced to remain cached, and I can't use the timestamp method, because I don't want ALL of the images to be reloaded every time the page loads.  I've racked my brain and scoured the Internet for a way to do this (preferrably via javascript), but no luck. Any suggestions? Switching languages in bibliography with biblatex I would like to have a bibliography with both Cyrillic and Latin and the main document language being English. How can I make biblatex to automatically switch between languages? (without using XeTeX as suggested by some) The minimal example returns an error if the main language is English. However, with the main language being Russian the document works.  \documentclass{article}  \usepackage[utf8]{inputenc} \usepackage[russian,english]{babel}  \usepackage[backend = biber,babel=other]{biblatex} \usepackage{csquotes} \addbibresource{library.bib}  \begin{document} %\selectlanguage{russian} %if the main document language is Russian the bibliography is compiled without any problem. This is a test. \parencite{Bara2006,Baranov2001}  \printbibliography  \end{document}   The .bib entries are  @book{Bara2006, address = {New York}, author = {Bara, Judith}, publisher = {Routledge}, title = {{English Citation entry}}, year = {2006} }  @book{Baranov2001, address = {Санкт-Петербугр}, author = {Баранов, Николай А}, title = {{Эволюция взглядов}}, year = {2001} } Is it ok to answer based someone's older answer on SE? Possible Duplicate:             This question will probably be closed in a few minutes. It is a duplicate but I couldn't find the question using with meta search.  Anyway, , . In this case, is vote to close the new question as a possible dublicate not the best choice? Looks to me a little bit not ethical. Is this correct?     That is a real cool answer.   I learned that that was incorrect, since "real" is an adjective which can describe a noun, e.g. "real answer" but it is not an adverb which can describe an adjective, "real cool". Instead you would have to say:     That is a really cool answer.   Since "really" is an adverb. Some observations I made is for $\frac{a}{b}+\frac{b}{a}$, is either:   the denominator has to be one, the numerator has to be a multiple of the denominator or  the numerator and denominator have to be the same.    Obviously, with the given conditions case 3 is eliminated since  $a \neq b$.  For case 1, if $b=1$, then $a=1$ which is a contradiction to the given condition that says $a \neq b$.  For case 2, I would think of examples. For example if $b=2$ then a multiple is $4$, so $a=4$. Then we have:  $$\frac{a}{b}+\frac{b}{a}=\frac{4}{2}+\frac{2}{4}=\frac{10}{4}$$  which is not an integer, but how would i proceed to show this case for all integers. How can I recover photo files from a memory card with a corrupt filesystem? I know this is not specifically a photography issue but since many of us with prosumer/professional-level DLSRs use Compact Flash cards, it is bound to happen to others as well.  I have a Sandisk 16GB Ultra card that occasionally had the camera reporting inability to access the card. I had to reinsert the card to continue using it. I was able to read the card and transfer the photo files to my computer drive. Today, the same thing happened again, just that nearing the capacity of the card, it failed one last time and the camera was subsequently unable to read it anymore. Despite reinsertion.  My other CF cards never once exhibited this problem, so it's pretty much isolated to this card.  The card is similarly unrecognisable on my computer CF card reader. I think the filesystem is toast, but I suspect the photo files are still intact. I would like to find ways where I can inspect the raw binary data in the card and hopefully recover the photos? Prove the map sending a matrix to it's inverse is continuous. Let $GL(n,\mathbb{R})$ be the space of invertible $n \times n$ matrices over $\mathbb{R}$. I have already proved that $GL(n,\mathbb{R})$ is open in $\mathbb{R}^{n^2}$ and that it is not connected with the relative topology from $\mathbb{R}^{n^2}$. How do I prove that the map $f:GL(n,\mathbb{R})\rightarrow GL(n,\mathbb{R})$ given by $f(x):=x^{-1}$ is continuous?  Any help would be greatly apprecaited.   Additionally how do I prove that $g:GL(n,\mathbb{R})\times GL(n,\mathbb{R}) \rightarrow GL(n,\mathbb{R})$ given by $g(x,y):=xy$ (matrix multiplication) is continuous, where we take the relative topology from $\mathbb{R}^{n^2} \times \mathbb{R}^{n^2} \cong \mathbb{R}^{2n^2}$? Solve for $f(x)$ if $f(f(x))=6x-f(x)$ Can you fly over a Medium enemy creature in a 10ft tall corridor? IllegalArgumentException or NullPointerException for a null parameter? I have a simple setter method for a property and null is not appropriate for this particular property. I have always been torn in this situation: should I throw an , or a ? From the javadocs, both seem appropriate. Is there some kind of an understood standard?  Or is this just one of those things that you should do whatever you prefer and both are really correct?