Let $A \subset \mathbb{R}^{n}$ be a bounded set and $f$ be a continuous function,  $ f:\mathbb{R}^{n}\to\mathbb{R}^{m}$. How do I show that $\overline{f(A)} \subset f(\overline{A})$?  What I have tried so far: $A \subset \overline{A}$, so $f(A) \subset f(\overline{A})$ and $\overline{f(A)} \subset \overline{f(\overline{A})}$. Since, $A$ is bounded we have a closed ball $B$ such that $A \subset B$. I'm not sure where to go from here. Any suggestions?
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?
When to use the double not (!!) operator in JavaScript Why is the t-test designed for small samples? Is it possible to flash a custom ROM without a custom recovery image For an un-rooted device, almost all articles/forum posts I've read say that in order to flash a custom ROM you first need to flash (or boot) a custom recovery image like TWRP.   I'm not sure I understand why. If the bootloader is unlocked shouldn't it be possible to manually flash images that make up the custom ROM to the required partitions with fastboot flash &lt;partition&gt; [ &lt;xxxx.img&gt; ], and leave the stock recovery untouched.  I realize that this may be difficult and that there is no benefit in doing it this way, but I'm curious to know whether or not it is possible, and if not, why? Hi I am working on a website for a student organization of my college.  In order to prevent spams, I have been looking for ways to limit account applicants only with email addresses from the college but haven't figure out how I'm trying to understand why the find command is not deleting files in one particular case. I've got the following command:  find C:\\path\\to\\Pictures\\pmcctv -name cap_*.jpg -o -name cap_*.ogg -o -name cap_*.flv -o -name cap_*.mp4 -o -name cap_*.webm -mtime +7   Which returns the files I would expect:  C:\path\to/Pictures/pmcctv/cap_20160915T193251_620067800.jpg C:\path\to/Pictures/pmcctv/cap_20160915T193322_742708800.jpg C:\path\to/Pictures/pmcctv/cap_20160916T081046_394767500.jpg C:\path\to/Pictures/pmcctv/cap_20160916T081125_615129600.jpg C:\path\to/Pictures/pmcctv/cap_20160916T081129_503678200.jpg C:\path\to/Pictures/pmcctv/cap_20160916T081255_842394600.jpg C:\path\to/Pictures/pmcctv/cap_20160916T161008_693586800.jpg C:\path\to/Pictures/pmcctv/cap_20160916T161012_749396800.jpg C:\path\to/Pictures/pmcctv/cap_20160916T161043_774132200.jpg C:\path\to/Pictures/pmcctv/cap_20160916T161150_497251900.jpg C:\path\to/Pictures/pmcctv/cap_20160916T161154_437379600.jpg C:\path\to/Pictures/pmcctv/cap_20160916T161337_350955300.jpg C:\path\to/Pictures/pmcctv/cap_20160916T161341_452596000.jpg C:\path\to/Pictures/pmcctv/cap_20160916T161412_870774000.jpg C:\path\to/Pictures/pmcctv/cap_20160916T161443_969064400.jpg C:\path\to/Pictures/pmcctv/cap_20160916T161554_468109900.jpg C:\path\to/Pictures/pmcctv/cap_20160916T161558_378086000.jpg C:\path\to/Pictures/pmcctv/cap_20160916T161916_668433100.jpg   Now if I try to run the same command with -delete, the files are not deleted:  find C:\\path\\to\\Pictures\\pmcctv -name cap_*.jpg -o -name cap_*.ogg -o -name cap_*.flv -o -name cap_*.mp4 -o -name cap_*.webm -mtime +7 -delete   I've also tried with -exec rm, but they are also not deleted:  find C:\\path\\to\\Pictures\\pmcctv -name cap_*.jpg -o -name cap_*.ogg -o -name cap_*.flv -o -name cap_*.mp4 -o -name cap_*.webm -mtime +7 -exec rm {} \;   find returns no error and the files are owned by me with permissions 644.  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Therefore  $$|f(A)|=|range(C_f)|\leq |A|=\aleph_0$$ $\therefore f(A)$ is a countable set. $\quad \blacksquare$  My problem is that I used the axiom of countable choice $AC_\omega$ in my proof but that isn't a part of the course I am taking. Does anyone have a simpler proof or one that does not involve Choice? Is a constant condition more costly than switching shaders? How can I remove a commit on GitHub? Winter is Coming It's my understanding that the phrase "Winter is Coming" is the motto of the Stark house, when the story first reaches the wall it's implied that the winter is not a periodic season like it is in our world.  Does the winter refer to a particularly severe season or does a winter only occur every few years? Installing AMD & Intel dual graphics Find a real function $f:\mathbb{R}\to\mathbb{R}$ such that $f(f(x)) = -x$? Change Recycling Bin Location I have a SSD as a primary hard drive, and one HDD for storage. I want the recycling bin to be on the HDD. 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