The problem:  Assume $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfies $|f(t) - f(x)| \leq |t - x|^2$ for all $t, x$. Prove $f$ is constant.  I believe I have some intuition about why this is the case; i.e. if $t$ and $x$ are very close, ($|t - x| = \epsilon$), then $\sqrt{\epsilon}$ will converge to 0, so $|f(t) - f(x)|$ will also converge to zero if you keep taking $t$ and $x$ closer and closer to each other. However, how do I formalize this argument? Thanks.
I want to show that if a function $f:[a,b]\rightarrow \mathbb R$ satisfies a Hölder condition of order $\alpha &gt; 1 $ then it is constant.  The way I think of it is as follows:  $$|f(x) - f(y)| &lt; K|x-y|^\alpha$$  $$\frac{|f(x) - f(y)|} {|x-y]}  &lt; K|x-y|^{\alpha -1}$$  $$\lim_{y\rightarrow x} \frac{|f(x) - f(y)|} {|x-y]} \le \lim_{y\rightarrow x}  K|x-y|^{\alpha -1} =0 $$ As the limit is $0$, we can remove the modulus, so we get:  $$\lim_{y\rightarrow x} \frac{f(x) - f(y)} {x-y} = 0$$ So $f$ is derivable and $f'(x) = 0$ for all $x$ in $[a,b]$. Note that the reason we can add the limit $y\rightarrow x$ is because $[a,b]$ is closed in $\mathbb R$.  However, the question gives as a hint using the mean value theorem. I am not sure why one should do that. You would first have to prove that $f$ is derivable in a similar manner to what I did, and then prove that $f$ is constant. Or is there a simpler way to do it and I am missing it?  Also please inform me of any mistakes I did in the proof (if any)/ Thank you!
Why can't ARIMA model large lags and/or long range dependence? ARIMA cannot model large lags (obtained from autocorrelation plot) and long range dependency (hurst exponent $H &gt; 0.5$). Why is it so? self referential struct definition? I haven't been writing C for very long, and so I'm not sure about how I should go about doing these sorts of recursive things... I would like each cell to contain another cell, but I get an error along the lines of "field 'child' has incomplete type". What's up?  typedef struct Cell {   int isParent;   Cell child; } Cell; I want to know all sets of single-variate functions that fulfill the following qualities: $\forall_{f:R^2 \to R^2} \forall_{g \in CAF} \forall_{h \in CAF} \exists_{k\in CAF} k(x) = f(g(x),h(x))$ where CAF is the set I am wishing to find. I know that the set of constants and the set of piece wise constants each satisfy that equation. Do any others? "They're using a cell phone" vs. "They're using cell phones" The usage of singular and plural has always been confusing for me.   I often see sentences like these         People are using cell phones.   People are using a cell phone.      Does the first sentence mean everyone has a phone and they are all using their own?  Does the second sentence mean they are sharing one cell phone?   If I see a group of people holding a cell phone in their hand(s), &lt;- even this is confusing for me, should I use the first sentence then?   Another example: you see two men, and both of them are carrying a bag. Which sentence should I say/use?        They are carrying backpacks          They are carrying a backpack      Could you please make it clear for me? Best way to merge mp4 files The origin of the value of speed of light in vacuum I would like to clarify the differences between by value and by reference.  I drew a picture     So, for passing by value,  a copy of an identical object is created with a different reference, and the local variable is assigned the new reference, so to point to the new copy  How to understand the words: " If the function modifies that value, the modifications appear also within the scope of the calling function for both passing by value and by reference "  Thanks! is it possible to display a Google Earth map INSIDE Excel? Is there any way, using the scripting available to Excel 2007 to display a Google Earth map inside a sheet?  I've used Excel to house the data that I've then gone and generated a KML file to load into Google Earth, but now they want to see a pretty little cell with a client address mapped out in Excel directly to pretty up the presentation. The Setup: Let's say we want to study a Euclidean $\mathrm{CFT}_2$ on $\mathbb R^2$ with coordinates $\sigma^1$ and $\sigma^2$ and metric   $ds^2 = (d\sigma^1)^2+(d\sigma^2)^2$.    It seems to me that in the usual discussion (e.g. di Francesco, Ginsparg, Polchinski), one proceeds to consider an analytic continuation of the CFT to $\mathbb C^2$ with coordinates $z^1, z^2$ and complex metric  $ds^2 = (dz^1)^2+(dz^2)^2$  and then, one performs the coordinate transformation $z = z^1+iz^2$ and $\bar z = z^1-iz^2$.  In this way the coordinates $z$ and $\bar z$ can be considered "independent" because they are coordinates on a complex two-dimensional manifold.  Also, in these coordinates the metric becomes  $ds^2 = dz\,d\bar z$  and it becomes clear that conformal mappings consist of mappings: $(z, \bar z)\to (f(z), g(\bar z))$.  My confusion is this: Since our original theory was on $\mathbb R^2$, books say that when we do calculations, we should consider the physical theory as living on the copy of $\mathbb R^2$ embedded in $\mathbb C^2$ given by the condition $\bar z = z^*$.  But consider the mapping $(z, \bar z)\to (z^2, \bar z)$.  This is a conformal mapping on $\mathbb C^2$, but it does not map the surface $\bar z = z^*$ to itself; for example the point $(z, \bar z)=(2,2)$ gets mapped to the point $(z^2, \bar z) =(4,2)$ and $2$ is clearly not equal to $4^*$.  In particular, it seems to me that analytic continuation to a CFT on $\mathbb C^2$ enlarges the set of mappings one can have, so what relevance does it really have to the original CFT on $\mathbb R^2$? Is there a way to force Google Photos to recognize new faces? I have "Face grouping" enabled in my Google account and I can see around 10 faces successfully tagged when I go to my Albums. However, there's a lot of other people who are not tagged, as well as some unrecognized photos of people who are already tagged.   Is there a way to force Google Photos to run the facial recognition algorithm on a given photo, so that it adds new faces or maps a new photo to an existing face? "Access denied" when trying to end a process on Windows 8 Side-by-side markdown title not so side-by-side In edit revisions where a long enough title has been edited, the side-by-side  markdown view shows a large overlap when the title wraps:    This is a repeat of  specific to the new Meta design. Looks like a line spacing adjustment is needed.  See this . Is there a word for the sadness over "What might have been"? Does this constitute a valid proof that $\frac{x^2}{1+x^4} \leq \frac{1}{2}$? Repost it with a minimal working example for the amsmath must be loaded earlier than mathspec problem The Android OS monitors the power consumption of running apps and hardware (display). As you may guess, this is a useful feature as it allows to track down energy greedy apps.  Is there an application that would do such monitoring on Mac OS?  It should calculate the percentages of the charge that an application/process/hardware has used.  I use plenty of little software that runs in the background permanently. It would be nice to know if such little helpers drain the battery significantly. Removing unnecessary vertices? I have over sampled polygons that I need to simplify, without changing the shape. basically, I want to reduce the number of vertices, by eliminating the ones that are useless and have the polygons represented with the significant vertices only. I used to do that with ET GeoWizards Plug-in, but I don't have a license anymore. My table has 500.000 features, water polygons.   Any ideas? Weird syslog order Have you ever seen wrong order in /var/log/syslog ? I think that it causes only crontab.  Jun 28 22:20:01 alex CRON[2327]: (root) CMD (something &gt; /dev/null) Jun 28 22:21:01 alex CRON[2700]: (root) CMD (something &gt; /dev/null) Jun 21 03:49:01 alex CRON[2753]: (root) CMD (something &gt; /dev/null) Jun 28 22:23:01 alex CRON[2754]: (root) CMD (something &gt; /dev/null) Jun 27 05:08:01 alex CRON[2813]: (root) CMD (something &gt; /dev/null) Jun 26 00:19:01 alex CRON[2816]: (root) CMD (something &gt; /dev/null) Jun 28 22:25:01 alex CRON[2816]: (munin) CMD (if [ -x /usr/bin/munin-cron ]; then /usr/bin/munin-cron; fi) Jun 28 22:25:01 alex CRON[2821]: (root) CMD (something &gt; /dev/null) Jun 28 22:27:01 alex CRON[3225]: (root) CMD (something &gt; /dev/null) Jun 28 22:28:01 alex CRON[3240]: (root) CMD (something &gt; /dev/null) Jun 28 22:30:01 alex CRON[3290]: (root) CMD (something &gt; /dev/null) Jun 26 12:03:01 alex CRON[3291]: message repeated 2 times: [ (root) CMD (something &gt; /dev/null)] Jun 28 22:30:01 alex CRON[3291]: (munin) CMD (if [ -x /usr/bin/munin-cron ]; then /usr/bin/munin-cron; fi) Jun 28 22:31:01 alex CRON[3659]: (root) CMD (something &gt; /dev/null) Jun 28 22:32:01 alex CRON[3754]: (root) CMD (something &gt; /dev/null) Jun 16 06:41:01 alex CRON[3846]: (root) CMD (something &gt; /dev/null) Jun 28 22:35:58 alex cron[909]: (CRON) INFO (pidfile fd = 3) Jun 28 22:35:58 alex cron[944]: (CRON) STARTUP (fork ok) Jun 28 22:35:58 alex cron[944]: (CRON) INFO (Running @reboot jobs) Jun 28 22:36:01 alex CRON[1428]: (root) CMD (something &gt; /dev/null) Jun 28 22:37:01 alex CRON[3050]: (root) CMD (something &gt; /dev/null) Jun 28 22:38:01 alex CRON[3082]: (root) CMD (something &gt; /dev/null) Jun 28 22:39:01 alex CRON[3114]: (root) CMD (something &gt; /dev/null)   As you can see, there is bad time order between lines. It is just crontab error, or there is some problem with system clock, or something?  I am running the latest Ubuntu 14.04 kernel:  Linux alex 3.13.0-30-generic #54-Ubuntu SMP Mon Jun 9 22:45:01 UTC 2014 x86_64 x86_64 x86_64 GNU/Linux I am trying to remove the Facebook and Messenger apps from my Asus Zenphone Laser 2.  Unfortunately, in both cases, I seem to be unable to find the option to fully delete them from the device. I have disabled them, but the app management screen now shows the following:     How can I fully delete/uninstall them? What is the square root of $i^4$? What is the $\sqrt{i^4}$?   $i^4$ = $(i^2)^2$ So is $\sqrt{i^4}$ = $\sqrt{(i^2)^2}$ = $i^2$ = $-1$?  Or is $\sqrt{i^4}$ = $\sqrt{1}$ = $1$?  When I plug it into my TI-89 Titanium, I get $1$.  Edit: I now see why $i^4 \neq -1$. $$\sqrt{x^2} = \left|{x}\right| $$ So$$\sqrt{(i^2)^2} = \left|i^2\right| = \left|-1\right| = 1$$