Let $X$ be a metric space, let $K$ be a subset of $X$, and let $f:K\to\mathbb{R}$ be a Lipschitz continuous map with some Lipschitz constant $L$ (thus $|f(x)-f(y)|\leq Ld(x,y)$ for all $x,y\in K$) Show that there exist an extension $\widetilde{f}:X\to\mathbb{R}$ with the same Lipschitz constant $L$.   I know that we can extend $f$ to the closure of $K$ with the same Lipschitz constant, thus we may assume $K$ is closed. If we then use Tietze extension theorem,  we can get of continuous extension $\widetilde{f}$ of $f$, but $\widetilde{f}$ may not be Lipschitz continuous with Lipschitz constant $L$. The hint says that a "greedy" algorithm will work here: pick $\widetilde{f}$ to be as large as one can get away with. I don't understand what the "greedy" algorithm is.
does anyone know the following result? If it holds in this form and any source which presents it? Thanks a lot.  Consider metric space $(X,d_{X})$.  Let $f:A \subset (X,d_{X}) \rightarrow \mathbb{R}$ be a Lipschitz continuous function i.e. $\exists K &gt; 0$ such that  \begin{align*} |f(x)-f(y)| \leq Kd_{X}(x,y) ~~~~\forall x,y \in A \end{align*}  then define functions  $$\bar{f_{1}}(x) := \sup\lbrace f(y) - Kd_{X}(x,y): y\in A \rbrace ~~ \forall x  \in X$$ and $$\bar{f_{2}}(x) := \inf\lbrace f(y) + Kd_{X}(x,y): y\in A \rbrace ~~ \forall x  \in X$$ it follows that  \begin{align*} \bar{f_{1}}, \bar{f_{2}}: X \rightarrow \mathbb{R}  \end{align*} are Lipschitz continuous with  \begin{align*} |\bar{f_{i}}(x)-\bar{f_{i}}(z)| \leq Kd_{X}(x,z) ~~~\forall x,z \in X \text{ and } \forall  i = 1,2 \end{align*} and  \begin{align*} \bar{f_{i}}(x) = f(x) ~~~ \forall x \in A \text{ and } \forall i =1,2 \end{align*}
Record a macro of windows commands Using Windows 7, is there an in-built or third-party method of recording windows commands and being able to play them back?  As a very basic example of what I would like to be able to do, imagine having two desktop themes that you like to alternate between. The Windows commands would be something like   Open personalisation applet select second theme ok   (I'm calling this a macro because of the equivalent capability in Word.) Cannot find Background Images feature to add in a photo in Blender 2.8  Where is it? Finding the index of an item in a list I asked a question a couple of days ago. I didn't receive any answers to the question, save for a few comments.  Once the bounty period was up I placed a bounty on the question as the concern at the time was rather urgent.  I got some answers to question eventually -- as I had expected. But requirements have changed since then and the content of the question has no longer become relevant.  What can or should I do with this question? Should I award the bounty anyway, regardless of whether the question still has it's relevance? How can I make Firefox download different file types to different locations? 98% of what I download is music, so I set the download location to my music folder. However, the other 2% has no place there, so every time I download .doc files or PDF files I have to remove them. Is there a way to make Firefox change the download folder based on file extension? I'll settle for all non-MP3 files getting saved to the desktop.  I know I can right click, choose Save Link As and pick the download location myself, but it seems like there should be a better way. Cannot install Ubuntu in VirtualBox due to "this kernel requires an x86-64 CPU, but only detects an i686 CPU, unable to boot" error I was trying to install Ubuntu 12.04 in VirtualBox 4.2.12r84980. 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I congratulated him for coming first in the race.      I congratulated him on coming first in the race. The ship flying away from Rey in her flashback of being left on Jakku looks like one of the ships in the group that takes her away when she is captured.  If so, and it is a First Order ship that dropped her off, this is an interesting Easter egg. Probability to obtain more than X with 3 dice. I love the  that was recently added onto Stack Exchange. However, this feature seems to be missing in per-site metas.  For instance:  Server Fault Meta:    DBA Meta    The parent websites, do have the edit/improve this post link however. Blank screen after installing nvidia restricted driver How do I select background music for my RPG campaign? Music is an essential component of my games. When running games set in the Star Wars universe, the music of John Williams adds a new dimension to the experience, both for me and for my players. Now that I'm starting a fantasy campaign, I'm a bit more uncertain about which music to use.  How do I create a soundtrack suitable for using in a fantasy campaign?  Specifically:   What sort of characteristics make for good back ground music? What makes for bad background music? What should I look for to get a fantasy feel? Which genres/sub-genres support fantasy games? What are good sources for fantasy background music?   A few notes:   Because the music will be used in the background of game sessions, most of the tracks should be one minute or longer, to avoid repetitive loops. Likewise, soundtracks with a variety of themes are preferred, so that they can easily support different modes of the game: exploration, combat, and chases, to name a few. Orchestral or acoustic music is preferred over music with vocals or electric instruments. I  have had no luck getting the buffer tool to accept anything but degrees as units of measure.  I have found lots of stuff saying the layer needs to be reprojected and saved but it hasn't worked at all for me.   Is there a way I could create a buffer without using ftools or at least force the units to meters somehow?  As a workaround I converted meters to degrees (lat) and used that but the final product needs to be as close to reality as possible.  Things I've tried:   setting every unit option I could find to meters (where possible). setting everything to NAD83/Maryland (data is for Washington, DC) and saving it as such (as layers in ESRI shape files). reimporting the reprojected layers setting relevant layers to Google Mercator   The was tried followed by creating a buffer. Many were tried in combination. QGIS 1.7.3 Slackware64 current (qgis from SBo-13.37 repo, tried on multilib and plain 64it with same results) Trying to find the original story (I read it in Russian translation, original was most likely English unless it was Lem).  The story plot had artificial birds designed to serve as cops. The birds started attacking humans who were percieved as being aggressive towards anything alive. So they created more advanced birds to protect against the first ones. This worked for a while until the second kind started attacking humans as well. It ended with the idea that they would need another generation to attack the second-generation birds.  The story wasn't very long. I read it a while back (mid-1980s, which means the original was written even earlier, probably before 1980).  Despite a somewhat Asimov-like "misbehaving robots" topic, I'm pretty sure it wasn't Asimov. Is there a database of Linux compatible hardware? I want to know if there is a centralised community site where all Hardware is listed in terms of compatability with Linux. Something like AppDB for Wine.