Find all continuous $f :[0,1]\to \mathbb R$ such that $\forall x\in [0,1], f(x)+f(x^2)=x$   I suspect there are none.  I made little progress so far, but it's worth noticing that $f(0)=0$ and $\displaystyle f(x)+\lim_{n\to\infty}\left((-1)^n\sum_0^{n-1}(-1)^{k+1}x^{2^k}\right)=0$
Prove that there is no continuous functions $f:[0,1]\to R$, such that   $$ f(x)+f(x^2)=x. $$   My try. Assume that there is a continuous function with this property. Thus, for any $n\ge 1$ and all $x\in [0,1]$, \begin{align*} f(x)&amp;=x-f(x^2)=x-\big(x^2-f(x^4)\big)=x-x^2+\big(x^4-f(x^8)\big)=\cdots\\ &amp;=x-x^2+x^4-\cdots+(-1)^n\left(x^{2^n}-f\big(x^{2^{n+1}}\big)\right) \end{align*}  since $f(0)=0$ and $\displaystyle\lim_{n\to \infty}x^{2^{n+1}}=0$ for any $x\in(0,1)$,it follows by the continuity of $f$ that $\displaystyle\lim_{n\to \infty}f\big(x^{2^{n+1}}\big)=0$, hence $$f(x)=x-x^2+x^4-x^8+\cdots+(-1)^nx^{2^n}+\cdots$$ for any $x\in (0,1)$  Why do I have prove that there is exists such a continuous functions $f$? Maybe  my example is wrong? Why?  if my   method is wrong,then How prove this problem ? Thank you.
Running Synaptic I get the following error message:  E: The package hl1440lpr needs to be reinstalled, but I can't find an archive for it. E: Internal error opening cache (1). Please report.   Upon accepting the message, Synaptic quits. Ubuntu16.04 boots into initramfs Ubuntu doesn't boot up and it goes into initramfs. I am writing from a live CD trial session. what should I do?  I understand that the boot flow is broken and that when the files from root are going to be loaded into the ram, it does not find the files in the root directory. Please correct me if I'm wrong.  The following is the output from ashell as I input the "exit" command:  (initramfs) exit /dev/mapper/ubuntu--vg-root contains a file system with errors, check forced. Inodes that were part of a corrupted orphan linked list found.  /dev/mapper/ubuntu-vg-root: UNEXPECTED INCONSISTENCY; Run fsck MANUALLY.     (i.e., without -a or -p options)  fsck exited with status code 4. The root file system on /dev/mapper/ubuntu--vg-root requires manual fsck How do I get label of the selected item from a lightning:select drop down list? I would like to get the value of the label of the selected drop down item.  In this case, the label is the Name of the Contact.  &lt;aura:component&gt;  &lt;aura:attribute name="contacts" type="Contact[]"/&gt;  &lt;lightning:select label="Contact" aura:id="contactList" name="contactList"&gt;     &lt;aura:iteration items="{!v.contacts}" var="c" indexVar="index"&gt;        &lt;option value="{!c.Id}"&gt;{!c.Name}&lt;/option&gt;    &lt;/aura:iteration&gt;  &lt;/lightning:select&gt;  &lt;/aura:component&gt;   I know how to get the value:  component.find("contactList").get("v.value");   But how do I get the label? Where in the registry does Windows store with which program to open certain file types? After installing Adobe PDF reader, it opens files with the .PDF file extension. Ones' default web browser opens .html files and notepad opens .txt files, etc.  Where exactly in the registry are these filetype - program associations stored in the registry?   In other words, how does Windows know with which software/program it must open a certain file(type)? Official StackExchange app says my question is not available? I recently posted a new question on Android Enthusiasts using the official StackExchange app for Android. Right after posting the question, the app said that my question is not available.     The question is not available, it may have been deleted.   Deleted? Already?...  Then I navigated to my user name to see my recent activity. There it was! I could see the question I just posted. But if I tapped on the link to try to open it, the app kept complaining about my question not being available.     The question is not available, it may have been deleted.   But 7 minutes after posting the question, I could see my question. Here are some screenshots that showcase this issue.      This is not the first time this happens. This appears to be more or less normal behavior of the app. I don't use the StackExchange app very often to post questions. But I had exactly the same message come up when I posted a question a few months ago (and some app version numbers earlier).  This is not a major issue, but you should try to address this nevertheless. It's a small bug, but addressing it might imply much bigger changes... which might explain why it's still present... I don't really know, I'm guessing. I just thought I would let you know about it, in case you are not aware of it.  For the record, the phone is Galaxy S4 Plus (GT-I9506) running Android 4.4.2. Version number of the app is 1.0.50. What is the best way of searching the whole classpath for an annotated class?  I'm doing a library and I want to allow the users to annotate their classes, so when the Web application starts I need to scan the whole classpath for certain annotation.  Do you know a library or a Java facility to do this?  Edit: I'm thinking about something like the new functionality for Java EE 5 Web Services or EJB's. You annotate your class with @WebService or @EJB and the system finds these classes while loading so they are accessible remotely. @ variables in Ruby on Rails New tags page mangles tag wiki excerpts css background color with floating elements How to align two images side by side if one label has more lines I have two same sized images and want them side by side with a caption under them.  My code looks like this:   \begin{figure}[H] \centering \begin{minipage}{.5\textwidth}   \centering     \includegraphics[width=\textwidth]{3}     \caption{label1}   \label{fig:test1} \end{minipage}% \begin{minipage}{.5\textwidth}   \centering     \includegraphics[width=\textwidth]{4}     \caption{label2}   \label{fig:test2} \end{minipage} \end{figure}   which works fine until one of the labels has 2 lines, than it looks like this:    and I woud like to prevent it to look something like this:    How can it be done?  Thanks Do the Faceless Men take more than just a face? Inequality vs. Inequation, Equation vs. Equality When talking about statements that two terms are mathematically (un-)equal (e.g., 1 = 1, 1 &lt; 2, or 1 != 2), what is the correct notion for such mathematical statements? Are there special cases w.r.t. "mixed" relations like x &lt;= y? Can some notions be used interchangeably?  It seems many online sources use "equation" and "inequality" but there is also a . Apart from personal posts in forums, I have not found any sources using "equality" in such a situation, yet. Systematically, I would think that "equation" and "inequation" would make more sense than "equation" and "inequality", but not everything (probably very few things) in natural languages have developed according to a purely logical system.  Note that this question is more general than  for that it particularly asks for the relation between the notion for equations and the one for inequalities. Equations are not part of the existing question at all. Should one ask generic or specific question? Let's assume that one has some problems with coding or designing specific project. Should one ask general question about techniques, algorithms, generic problem (anonymising the project references), or should one ask about specific problem in specific problem? Or perhaps something in between?  Example:   generic: Cross-browser implementation of “HTTP Streaming” (push) AJAX pattern specific: How to implement AJAX-y 'incremental blame' view in gitweb (git web interface in Perl)?   If one should ask generic questions, how to anonymise problem? If one should ask specific question, how avoid 'homework' syndrome? How to plot an emission spectrum? Where can I find sample or real Selenium projects. All the material I can find on github and on internet are basic, and never intermediate or advanced.  How does a real Selenium project look in real life? How can I find sample code? Why is HMAC-SHA1 still considered secure? This Q &amp; A   says that the security of HMAC-SHA1 does not depend on resistance to collisions? Are they are saying specifically with respect to HOTP or for HMAC-SHA1 for any use?     If HMAC-SHA1 is used for verifying the integrity of a document and you can find a collision, you can make changes in the document, right?      Even the RFC says that resistance to collisions is relevant       The security of the message authentication mechanism presented here depends on cryptographic properties of the hash function H: the resistance to collision finding (limited to the case where the initial value is secret and random, and where the output of the function is not explicitly available to the attacker), and the message authentication property of the compression function of H when applied to single blocks    What am I missing here? Define $\mathcal N$  $\mathcal N (A):=\mathcal P(A)\setminus\{\varnothing\}$  Does $\mathcal N$ has a special name and standard notation? I was editing my mesh and I wanted to only see a part of my mesh, so I pressed alt+B to only show a selected area. However, I did ctrl+B and created a red rectangle on my screen (I don't know what it does). Does anybody know how to get it off? Are the LIGO observations proof of a black hole merger, and what happened to the black holes? Richard Stallman states that Ubuntu "has installed surveillance code". See:   Is this true?   I agree with him that "Any excuse Canonical offers is inadequate....".