$f:I \rightarrow \mathbb R$ is said to be Hölder continuous if $\exists \alpha&gt;0$ such that $|f(x)-f(y)| \leq M|x-y|^\alpha$, $ \forall x,y \in I$, $0&lt;\alpha\leq1$. Prove that $f$ Hölder continuous $\Rightarrow$ $f$ uniformly continuous and if $\alpha&gt;1$, then f is constant.  In order to prove that $f$ Hölder continuous $\Rightarrow$ $f$ uniformly continuous, it is enough to note that $|f(x)-f(y)| \leq M |x-y|^\alpha \leq M|x-y|$, since $\alpha \leq 1$. This implies that f is Lipschitz $\Rightarrow$ f is uniformly continuous.  But how can I prove that if $\alpha &gt;1$, then $f$ is constant?
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!
Does the hom-tensor adjunction hold for $O_X$ modules also? With sheaf hom and sheaf tensor product, the statement would consist of a natural transformation $Hom_O (M \otimes_O N, K)\cong_{nat} Hom_O(M, Hom_O(N, K))$, $O = O_X$ is the structure sheaf and $M,N,K$ are sheaves of $O_X$ modules.  If true I think I can check this by hand, by describing sheaves in terms of compatible germs, defining this adjunction on the germs and checking that they glue together. It is easier in the quasi-coherent sheaves on a scheme case because then one can just work in an affine cover and the distinguished base.  I am little bit concerned that one might need a finite presentation hypothesis somewhere, in order for $Hom$ to localize well. ($Hom_{R[S^{-1}]}(M[S^{-1}], N[S^{-1}]) \cong Hom_R(M,N)[S^{-1}]$ needs $M$ finitely presented.) What is the difference between a "link function" and a "canonical link function" for GLM What's the difference between terms 'link function' and 'canonical link function'? Also, are there any (theoretical) advantages of using one over the other?  For example, a binary response variable can be modeled using many link functions such as logit, probit, etc. But, logit here is considered the "canonical" link function. I have a Asus ROG G551VW laptop and I'm trying to install Ubuntu 16.04 on my laptop, but when i try to load the installer, my laptop just freezes and its fan starts working for no reason. I have tried to install Ubuntu 14.04 on my laptop and the installation went smoothly. What should I do if I want to install Ubuntu 16.04 on my laptop?  Any help would be greatly appreciated. I seem to remember from first year physics that we can calculate the RMS speed of a stationary, ideal gas with $v=\sqrt{\frac{3RT}{M}}$. Does a similar equation exist for liquids? Don't worry, I have written your name on the attendance sheet.   Or     Don't worry, I wrote your name on the attendance sheet. What are the different colors on the unity launcher icons? Example:      My question is simple.  What are the different colors representing in the unity launcher background for each icon.  I mean, it seems consistent (firefox is always orange), but it's not jumping out at me what the pattern is.  On my screen, it's:   Firefox: Orange Chrome: Yellow Terminal: Grey RDP App: Blue KeyPassX: Green NetBeans: Blue   I'd like to turn it off if possible, but first I'd like to understand more about it.  Any launcher experts? Boot stuck at logo screen after upgrading to 14.10 I am trying to help my daughter on her math homework and I am having some trouble on some equation solving steps. My current major concern relies on understanding why $\sqrt{8}/2$ equal to $\sqrt{2}$.  Thanks in advance Man undergoes radical surgery to research aliens Can I see which seats are available on a British Airways flight before I book? different result for yyyy-mm-dd and yyyy/mm/dd in javascript when passed to "new Date" In the movie Harry Potter and the Goblet of Fire, there's a scene where the students are preparing for the Yule Ball.   Professor McGonagall is instructing the boys and girls, while Filch is choosing a vinyl record to place in an enormous turntable phonograph with a horn as big as Hagrid.   This has me wondering if the wizarding world, in general, plays vinyl records. Or, if this was just a Muggle artifact that Hogwarts kept as a novelty.   Are there other instances of witches or wizards using or owning phonographs or record players in either the books or movies? If so, what are they? (Examples help to illustrate if this was a one-off or a trend) How to show that this binomial sum satisfies the Fibonacci relation? The binomial sum   $$s_n=\binom{n+1}{0}+\binom{n}{1}+\binom{n-1}{2}+\cdots$$   satisfies the Fibonacci relation.   I failed to prove that $\binom{n-k+1}{k}=\binom{n-k}{k}+\binom{n-k-1}{k}$... Any hints or suggestions? Do the glowing pillars in The Storm Halls do anything? I've noticed that occasionally if I shoot at (or near) the glowing red pillars they will kind of explode a bit for a second and then reform. Does this affect my attacks in any way? What about nearby enemies? Is this purely cosmetic (like pillars in other dungeons crashing down)? What is the best way to call a script from another script? An inequality for arctan$x$ Why it is true that $|\arctan x-\arctan y|\leq |x-y|$? I have tried to use $\tan x$ but I didn't conclude to something. $A \oplus B = A \oplus C$ imply $B = C$? I don't quite yet understand how $\oplus$ (xor) works yet. I know that fundamentally in terms of truth tables it means only 1 value(p or q) can be true, but not both. But when it comes to solving problems with them or proving equalities I have no idea how to use $\oplus$.  For example: I'm trying to do a problem in which I have to prove or disprove with a counterexample whether or not $A \oplus B = A \oplus C$  implies $B = C$ is true.   I know that the venn diagram of $\oplus$ in this case includes the regions of A and B excluding the areas they overlap. And similarly it includes regions of A and C but not the areas they overlap. It would look something like this:    I feel the statement above would be true just by looking at the venn diagram since the area ABC is included in the $\oplus$, but I'm not sure if that's an adequate enough proof.  On the other hand, I could be completely wrong about my reasoning.    Also just for clarity's sake: Would $A\cup B = A \cup C$ and $A \cap B = A \cap C$ be proven in a similar way to show whether or not the conditions imply $B = C$? A counterexample/ proof of this would be appreciated as well. I have installed a package using snap and I need to modify one of the files but when I try to change its ownership or permissions, I always get the following message:  sudo chmod +x ./my_file.js chmod: changing permissions of '/snap/my_app/my_file.js': Read-only file system   How can I modify files that are installed via snap? After restarting my computer when I am presented with the initial login screen (the light gray one with the restart and shut down buttons), I see only two of my three user accounts.  If I log in with one of these accounts, I can see, and log in to, the third (missing) account by either:   Logging off (Apple > Log Out...) to reach the second login screen (the dark gray one with sleep/restart/shut down buttons) Clicking on the fast user switching menu.   How can I restore this account in the initial (light gray) login screen?  I tried repairing disk permissions to no avail.  If I create a new account, it does show up in the list. utf8 and ISO 8859-1 encoding references trouble I have to use one particular template ( am obliged to use it for my thesis) it uses \bibliographystyle{plain} Ihave both english and frensh text on it (for abstract and résumé) there is no problem for characters like ( é, ç...) but some of references does not appear they are replaced by [?] knowing that foe encoding \RequirePackage[latin1]{inputenc} how to fix this problem please!   \documentclass{article}   \RequirePackage[latin1]{inputenc}   \RequirePackage{fancyhdr}     blabla \cite{nayak2015comprehensive, byun2002applications, shah2004sequential, fletcher2009support, cornuejols2011apprentissage, byun2002applications}   @article{nayak2015comprehensive,    title={A comprehensive survey on support vector machine in data mining     tasks: applications \&amp; challenges},    author={Nayak, Janmenjoy and Naik, Bighnaraj and Behera, HS},    journal={International Journal of Database Theory and Application},    volume={8},    number={1},    pages={169--186},    year={2015}    }      @inproceedings{shah2004sequential,     title={Sequential k-nearest neighbor pattern recognition for usable      speech classification},     author={Shah, Jashmin K and Smolenski, Brett Y and Yantorno, Robert E      and Iyer, Ananth N},     booktitle={Signal Processing Conference, 2004 12th European},     pages={741--744},     year={2004},     organization={IEEE}     }      @misc{fletcher2009support,      title  = {Support vector machines explained},     author = {Fletcher, Tristan},     journal= {University College London, London},     year   = {2009},     howpublished = "\url{http://sutikno.blog.undip.ac.id/files/2011/11/SVM-     Explained.pdf}",     note   = {Accessed: 2017-04-30}     }      @inbook{cornuejols2011apprentissage,      title={Apprentissage artificiel: concepts et algorithmes},      author={Cornu{\'e}jols, Antoine and Miclet, Laurent},      year={2011},      pages={279--309},      publisher={Editions Eyrolles}       }      @incollection{byun2002applications,      title={Applications of support vector machines for pattern recognition:       A survey},      author={Byun, Hyeran and Lee, Seong-Whan},      booktitle={Pattern recognition with support vector machines},       pages={213--236},      year={2002},      publisher={Springer}      }    \end{document}  knowing that my references work well on other template but this one no how to fix this problm please! here is the error I got   Edited here is the link to the template  in fact I have created a littele example ( since I could not publish my thesis ...) the error has disappeared but my references still [?]