Volume of a square pyramid-- what's wrong in my analysis
Triple integral limits over pyramid
Question: Let $a, b \in \mathbb{R}$ with $a &lt; b$ and let $f: [a,b] \rightarrow [a,b]$ continuous. Show: $f$ has a fixed point, that is, there is an $x \in [a,b]$ with $f(x)=x$.  I suppose this has to do with the basic definition of continuity. The definition I am using is that $f$ is continuous at $a$ if $\displaystyle \lim_{x \to a} f(x)$ exists and if $\displaystyle \lim_{x \to a} f(x) = f(a)$. I must not be understanding it, since I am not sure how to begin showing this... Should I be trying to show that $x$ is both greater than or equal to and less than or equal to $\displaystyle \lim_{x \to a} f(x)$ ? Are there specific techniques/filters for snowscapes? Why am I forced to use the newest chat interface? When logging into chat-stackexange I am presented with this message:  The new mobile version of chat is now the default mobile version on devices that support it. Unlike the classic (okay, ancient) version, it allows you to perform actions like editing, starring, and direct replies to particular messages. The classic version is still available and will be used in browsers that lack support for some necessary features of the new version. Logged in users can also choose to continue using the old version on the preferences page of their user profile. Please report bugs and other issues on Meta Stack Exchange. Thank you!  I choose to continue using the old version but cannot find the option to do so on my chat user profile page? In Harry Potter and the Philosopher's Stone, Hermione is in the toilets when Harry and Ron lock in the troll. She lies and says that she went looking for the troll. Why? She could have just said that she was in there because she needed the toilet (she was allowed to be, it's a girls' bathroom) and Harry and Ron came to save her.   Is there a reason she lied? Why would an incorrect password attempt take a lot longer to process than a correct one? The most prominent place I've noticed this is when SSH-ing at work, but I feel like I've observed this behaviour elsewhere too.  When I try to log into Linux servers from my Windows desktop at work, I've noticed that if I mis-type my password, it takes a good 5 seconds or so before I get "Access Denied" back. Then when I type my password correctly, the login (along with welcome messages etc) is virtually instant.  Is there any logical reason for this, or would it be down to some odd configuration that's specific to the machines here at work? What are the most downvoted questions and answers on Stack Overflow, Server Fault, Meta, and Super User history to date?  Similarly, what are the most upvoted questions and answers on SO, SF, Meta, and SU to date? Should I really have been awarded mortarboard on a Per-site meta? Discuss convergence of the series $\sum \frac{(1+n)^n}{n!}x^n$. Let    $$a_n = \dfrac{(1+n)^n}{n!}$$   Discuss convergence of the series $\displaystyle \sum a_nx^n$.   I've shown that the series converges absolutely when |x| &lt; 1/e and diverged when |x| > 1/e. I can't seem to determine what happens at |x| = 1/e Mount ext4 partition with >4KiB block size I'm attempting to mount a partition from a NAS and the block size is 65536 bytes, this is above my distribution's precompiled kernel memory paging size of 4KiB, so I cannot mount it normally.  Which kernel compilation parameters would I need to change to get a memory paging size large enough?  Would it be possible under the x86-64 architecture?  EDIT: There doesn't seem to be an option for it in make xconfig unless I'm just missing it.  From page_types.h under arch\x86\include\asm  /* PAGE_SHIFT determines the page size */ #define PAGE_SHIFT  12 #define PAGE_SIZE   (_AC(1,UL) &lt;&lt; PAGE_SHIFT) #define PAGE_MASK   (~(PAGE_SIZE-1))`   It doesn't look to be configurable.  I could change this number directly to 16 and it would probably make pretty explosions and destroy Tokyo.  I shall probably try it later this evening. Rename command not found when running a script In a batch file I need to extract a month, day, year from the date command.  So I used the following, which essentially parses the Date command to extract its sub strings into a variable:  set Day=%Date:~3,2% set Mth=%Date:~0,2% set Yr=%Date:~6,4%   This is all great, but if I deploy this batch file to a machine with a different regional/country settings, it fails because month, day and year are in different locations.   How can I extract month, day and year regardless of the date format? Set Windows process (or user) memory limit Is there any way to set a system wide memory limit a process can use in Windows XP? I have a couple of unstable apps which do work ok for most of the time but can hit a bug which results in eating whole memory in a matter of seconds (or at least I suppose that's it). This results in a hard reset as Windows becomes totally unresponsive and I lose my work.  I would like to be able to do something like the /etc/limits on Linux - setting M90, for instance (to set 90% max memory for a single user to allocate). So the system gets the remaining 10% no matter what. Identifying Coordinate System of Shapefile when Unknown? I have a Shapefile but its coordinate system is Unknown, and there is no *.prj file.   How can I identify it now? Can I use mirroring to make a copy of my Database? I am lost in normalizing, could anyone guide me please.  I have a minimum and maximum values, say -23.89 and 7.54990767, respectively.  If I get a value of 5.6878 how can I scale this value on a scale of 0 to 1. "Ghost" UI elements Occasionally, I get part of an application lingering behind on my desktop:    This is part of a context menu for an application. It is always top-most, meaning that it is always on the screen, and I cannot click on it (it just clicks through it). It has happened before, sometimes with whole windows. It remains on screen even if the application it originated from is no longer running.  How do I get rid of it without rebooting? This is an idea I have seen around, but I have not seen a feature request for it, so here it is.  Downvotes that have no comments are frustrating and difficult to manage.  This is a way to help improve that:   Have a downvote initially cost more to cast for the downvoter (5 points for example) When a comment (from the downvoter) on the downvoted question/answer gets an upvote, you get your points back.   All points (ie all 5) could be refunded or just up till the traditional -1 that is in place now (ie 4 points refunded).    This would encourage good comments and downvotes for real reasons.  (Because comments for questions/answers that are bias related will not get the upvotes as readily.)  This could/should be limited to non-subjective tags.  (ie tags where a downvote just because you "don't like it" is not appropriate.) Analytics search keywords are unreliable Command/macro name cannot include numbers and symbols I would like put picture into table, and my idea is define new command as below>   \newcommand{\EJ471}{\includegraphics[scale=0.150]{EJ_471.jpg}}   But compiler reports an error. If I use the command name for example \EJa, everything is all right. I've read similar questions as      But still I don't understand why it does not work. How to split a string in sql server 2008 using stored procedure and insert the data to table