Find a function that matches the following conditions. Find a function that matches the following conditions.  (a) $f(x)$ is continuous for all real numbers  (b) $f(0)$ = 3  (c) For all real numbers $x$, $f(x) = f(x/2)$  This is from a past paper, and the answer has been given as the following steps:   $f(a) = ... = f(a/2n)$ $f(a)$ = lim as n approaches infintiy $f(a)$ = lim as n approaches infinity of $f(a/2n)$ $f$(lim as n approaches infinity of $\frac{a}{2n}$ therefore $f$ is continuous $f(0) = 3$   My question arises due to poor printing of my answer book. In step 1 is $f(a/2n)$ the correct step or should it be $(f(\frac{a}{2^n})$? If so could you please explain why? Because for both possibilities I can't seem to figure out why that's a step in proving the continuity. I'm sorry for the very elementary question. Thank you!
If $f: \Bbb{R}\rightarrow\Bbb{R}$ is continuous at $0$ and $f(x)=f(2x)$ for each $x\in\Bbb{R}$ then $f$ is constant. How do I prove that if $f: \Bbb{R}\rightarrow\Bbb{R}$  is continuous at $0$ and $$f(x)=f(2x)$$ for each $x\in\Bbb{R}$, then $f$ is constant?
On Chrome on Android, code blocks on the mobile site have no syntax highlighting, are not a monospaced font, and words are wrapped.  Mobile site on mobile browser:    Desktop site on mobile browser:    Inline code has the same issue, it isn't a monospaced font.  It has been this way for as long as I can remember. Wouldn't be surprised if this is a dupe. Remove impossible badges from per-site metas Per-site metas lists some badges that are impossible to get. For example  lists:         But these are impossibly to get because per-site metas do not support bounties, were not in a beta stage or Area 51 proposals. I therefore suggest that these badges are removed from the per-site metas. They have no function and they are confusing. There are already badges that are unavailable on per-site metas but available on the main sites:    Convention Quorum  This seems to suggest that per-site metas should only contain badges that can be possibly awarded. Whenever I startup from the LinuxLive USB, and try to install Ubuntu 12.04, it only gives me two options instead of the usual 3 or 4. My options are "Replace Windows 7 with Ubuntu" and "Something Else". Whenever I go into "Something else", it looks extremely confusing. How do I install it side-by-side while using the "Something else" section? These are my partitions.   /dev/sda /dev/sda1 ntfs (104Mb, System Required Partition) /dev/sda2 ntfs (1000097Mb, Windows 7 Partition)   Or should I try reinstalling my installer on a CD rather than USB and maybe then it will have the "Install alongside" function? Microdisplacements with adaptive subdivision turning black What does Redis do when it runs out of memory? Array with labeling columns Why is angular velocity of any point about any other point of a rigid body always the same? I have the following question regarding an ideal rigid body.  Firstly, is it always true without any exceptions that the angular velocity of any point about any other point on a rigid body is always the same?  If so, what happens in case of precession, in which two different angular velocities are predefined? I have heard rumors that you can compile documents online, and more specifically that Google has a free online compiler, but I have never been able to find any. Is there a way to compile documents online, so that I can write and compile documents even if I don't have a TeX distribution installed on my computer? It’s a story of a female protagonist who has a harsh life. Just as she is about to die, she time travels back into her young self. To avoid the future she already lived, she decides to gets strong by learning swordsmanship. After a while, she joins the army under the command of a prince who wears an eye patch. She is quite OP, and helps the prince. It was kind of a medieval setting manga, with dukes and earls and count types. Time is when the use of sword and magic is prominent. Why does the Replace() string method not modify my string variable? How can I kill minerd malware on an AWS EC2 instance? (compromised server) I found malware on my ec2 instance which was continuously mining bitcoin and using my instance processing power. I successfully identified the process, but was unable to remove and kill it.  I ran this command  watch "ps aux | sort -nrk 3,3 | head -n 5"  It shows the top five process running on my instance, from which I found there is a process name 'bashd' which was consuming 30% of cpu.  The process is   bashd -a cryptonight -o stratum+tcp://get.bi-chi.com:3333 -u 47EAoaBc5TWDZKVaAYvQ7Y4ZfoJMFathAR882gabJ43wHEfxEp81vfJ3J3j6FQGJxJNQTAwvmJYS2Ei8dbkKcwfPFst8FhG -p x   I killed this process by using the kill -9 process_id command. After 5 seconds, the process started again. What is the difference in this specific breadboard wiring? Hyperspace travel is a norm in Star Wars universe. Even small ships have that capability and without this capability galactic-scale economy isn't possible.  My educated estimates say that at any point of time at least billions of ships are in hyperspace across the galaxy. Now, talk about economic hubs like Coruscant during Clone Wars Era. The number of ships arriving here should be enormous. Millions in an hour? If one person small ship can have warping capability, then the number can exceed even billions in a peak hour. You can go shopping on a neighbor planet five times a day or you can live on a cheap planet even if you have a job on Coruscant (there are public transport cabs featured in Rebels TV show in case you aren't rich enough to have own ship).  Now, the famous quote:     "Traveling through hyperspace ain't like dusting crops, boy! Without precise calculations we could fly right through a star or bounce too close to a supernova, and that'd end your trip real quick, wouldn't it?"   That's fine. Calculations prevent you from hitting a star, supernova or planet which are already on map, but how does a ship avoid hitting another ship? Provided that number of ships arriving near a planet is that much high, the probability of two ships jumping to the same co-ordinate can't be zero. In fact, the probability should be very very high.  Is there a central body which controls the traffic by coordinating the hyperspace calculations of different ships? This seems unlikely because everyone seems to go anywhere they want while maintaining the destination privacy (there are countless examples). Or, is there really a traffic control body and those who travel privately actually gamble? Is there any other protocol if not traffic control body?  Has the canon ever addressed it? What about Legends? Vertical placing of wide accents in superscript The following code  $\widetilde{aaa} \quad L^{\widetilde{aaa}}$  $\tilde{aaa} \quad L^{\tilde{aaa}}$  $\widehat{aaa} \quad L^{\widehat{aaa}}$  $\hat{aaa} \quad L^{\hat{aaa}}$  $\overline{aaa} \quad L^{\overline{aaa}}$  $\bar{aaa} \quad L^{\bar{aaa}}$   produces the following output:    What is the reason for additional vertical spacing for \widetidle and \widehat in superscript and what can one do about it? Continuously differentiable map from $\mathbb{R}^{m+n}$ to $\mathbb{R}^n$ Suppose $m,n&gt;0$, $U$ an open subset of $\mathbb{R}^{m+n}$ and let $f: U \to \mathbb{R}^n$ be continuously differentiable.  Is it possible for $f$ to be injective?    My thinking is that continuous differentiability+injectivity suggests that this is sort of getting at a converse to the Inverse Function Theorem.  We know the Jacobian is not invertible at any point in $U$, and the question is asking if this implies that $f$ cannot be invertible in any neighborhood contained in $U$. London Heathrow airport - do I need a transit visa? Python 3 with QGIS on Mac? I need to start learning Python, and I want to learn with version 3.x. I understand that QGIS--at least some plug-ins--relies on Python and Tcl/Tk, and probably on specific versions. Assuming that QGIS is dependent on Python 2, can anyone tell me how to set up parallel installations of Python 3 and Tcl/Tk such that QGIS will have what it needs and I can use Python 3 for other purposes? Irrationality proofs not by contradiction Per now, I have basically come upon proofs of the irrationality of $\sqrt{2}$ (and so on) and the proof of the irrationality of $e$. However, both proofs were by contradiction.  When thinking about it, it seems like the definition of irrationality itself demands proofs by contradiction. An irrational number is a number that is not a rational number. It seems then that if we were to find direct irrationality proofs, this would rely on some equivalent definition of irrational numbers, not involving rational numbers themselves.   Are there any irrationality proofs not using contradiction? I wanted to take a sneak peek at Windows 7 so I thought of installing it. I have Windows Vista on my laptop which came pre-installed with it. I didn't want to mess with it.  So I created a partition (20 GB) in my USB external hard disk and tried to install Windows 7 on that partition. But when I booted from a Windows 7 DVD and selected the target partition on USB hard disk, it said it cannot be installed.  Is there any way to install Windows on an external USB hard disk? Why would a static nested interface be used in Java?