if $f: \mathbb{R}\to \mathbb{R}$ satisfies $$|f(x)-f(y)|\le |x-y|^{\sqrt{2}}$$ for all $x,y\in \mathbb{R}$ ,then is f increasing ,decreasing or constant? in my view ,it is clear that $|f(x)-f(y)|$ is not zero so it is not constant. And since the rate of change is positive so the function would be increasing.Is my solution correct or wrong?
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!
How can I programmatically delete a line from a Word document using c#? How do I install Guest Additions in a VirtualBox VM? I've recently installed Ubuntu on Oracle's Virtualbox.  Now I want to install the guest additions.  I started up my Ubuntu guest OS, clicked "Devices" and chose "Install Guest Additions". Nothing happened, and I don't see the CD icon on my desktop.  How can I make this work? In "Tasksel" how do I select a package? Why does the integral of the surface area of a sphere equal it's volume?   $$\int{4\pi r^2 \ \mathrm{d}r =\frac{4}{3}\pi r^3}$$  I don't quite understand why the relationship between surface area and volume is this way. Is this true for any other shapes? I've installed reCAPTCHA module but spam still bypass my security and leave nasty comments. What should I do? How to block them? Downloading US Census data? When I spoke with a friend of mine, I had noticed she used sentences similar to "I need to go buy food."   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The switch is on the other side of the wall and it would be eextremely difficult and ugly to run another wire to it. (no crawl or attic space). So my question is, can I change the current outlet/wiring within the outlet to make it a half-hot? My understanding is that removing the brass connector will just make half of it dead. SciFi movie (or miniseries) of the afterlife I am trying to find the name of a direct to TV movie, possibly a 2-3 part miniseries which I believe aired on the SciFi channel in the 90s. I don't remember who starred in it, but the movie opened with an astronaut attempting reentry whose shuttle ends up exploding. He awakes in a large bubble of air in a river and after reaching the surface and swimming to shore, meets several other people, from seemingly many different times throughout history (including one alien, who died on Earth).  Over the course of the movie, we learn the world is the afterlife, where everyone goes when they die. The major plot involves launching a steamboat (built by Mark Twain, no less) to travel up a river to the mountains, where the beings responsible for bringing everyone to this world are.  Does anyone know what the title of this movie/miniseries is? Setting Boundaries around city in google maps Is the Macbook 12 inch's USB-C charger interchangeable with other USB-C chargers or devices? If there is only the Macbook 12 inch USB-C charger nearby, can it be used to charge other USB-C devices, such as any Android phones that also use USB-C? (will it be "too powerful" and damage other devices?)  Vice versa, if there is a Quick Charge 2.0 or 3.0 USB-C charger from other devices nearby, can it be used to charge the Macbook 12 inch?  The main concern is about whether it could damage the device or battery. I have this strange problem with conky. It doesn't show properly on the desktop:    I've tried everything from editing conky widget file to reinstalling conky and conky manager (v.2.4). Who knows how to fix it? first of all thank you for answering.  I have a 120GB SSD and a 1TB HDD and I want to dual boot with Windows 10. I tried with this configuration:  SSD  / (ext4 20GB) /usr (ext4 15GB) /boot/efi (ext4 250MB)  HDD  /var (ext4 3GB) /tmp (ext4 8GB) /home (ext4 50GB) swap (none 24GB) (RAM 16GB)  I don't know if I am doing OK, as I tried to install it but installation gets stuck at "Saving installed packages..." and it doesn't continue. I wonder if it could be cause I didn't  partitioned as I had to. I don't find any post where this happens so I ask here.  Thanks! Extracting embedded images from a PDF Before I started using Ubuntu I used Nitro PDF reader to automatically extract images from PDF files. Is there a PDF reader for Linux that does this?  I would like to be able to extract images faster/easier than when taking a snapshot. Propose Newbie-Overflow site or section Because the perl section is looking like that right now. I believe that Stack Overflow is going out of it's charter in becoming a newbie FAQ when as far as the original reasoning it was meant to be something along the lines of an expert site.   is a good example of what I mean in Perl. It probably could be closed because it's a duplicate, or some other reason.   I would also like to indicate that I don't hold back answering newbie questions. It's just when I think of an obscure "stack overflow" error that's beating you some 3 am close to deadline, that's not "Howcum Perl print ..." and follows with something that causes a vast majority of Perl programmers to say oops, because they know exactly what's going on.   So the proposal is   A new site. "Move to Newbieville" A new "section" A new status somewhat similar to Closed ("Newbied") but still can be updated all the same as an open question.   I'm not against giving some newbies an assist, but I feel a FAQ for rank newbieism is not something that SO started out to be, based on the original concept of a Expert Site.   Including comment to Gnoupi:   I said nothing about making newbies unwelcome. Like I said. I answer them--heck, I think I've even answer homework questions. I've answered on the vague impressions that I can glean from posts that have 4 close votes. It can even lead to avoiding downvoting or yelling at newbies by more established curmudgeons if there is a special action you can do to gently inform newbies that this is a real newbie question.  However, you have to know that to some people newbies are already unwelcome. There is a site ethic, but the individual is also going to decide how much they welcome newbies. And people do post RTFM answers already.   Anyway, I'm one vote away from a peer pressure badge it seems. (Nope, nailed it.)    EDIT  For somebody jumping to the conclusion that I'm hostile to newbies:  just earlier Resistor circuit that isn't parallel or series How to count the number of set bits in a 32-bit integer? How to run firefox and firefox ESR simultaneously