Let $\mathbb R$ be the set of real numbers and $f: \mathbb R \rightarrow \mathbb R$ be such that for all $x$ and $y$ in $\mathbb{R}$, $$|f(x)-f(y) |\leq |x-y|^3.$$ Prove that $f(x)$ is a constant.    This is a new type of problem for me and I feel I am missing some trick to simplify the given expression. Any help??
Suppose $f:(a,b) \to \mathbb{R} $ satisfy $|f(x) - f(y) | \le M |x-y|^\alpha$ for some  $\alpha &gt;1$ and all $x,y \in (a,b) $. Prove that $f$ is constant on $(a,b)$.   I'm not sure which theorem should I look to prove this question. Can you guys give me a bit of hint? First of all how to prove some function $f(x)$ is constant on $(a,b)$? Just show $f'(x) = 0$?
Grep: unexpected results when searching for words in heading from man page Customizing titlesec-etoc (further question) Clicking to reload an updated answer removes the code snippet buttons and removes syntax highlighting If I push or hit an object in space will it rotate or move along a straight line? If I push or hit an object in space (vacuum and no gravitation) in direction what is not going trough its centroid, will it rotate or move along in straight line?   I expect that on earth it will depend on what is less difficult for the object (rotation or linear movement). So the object will do some kind of combination of both movements (rotating and also moving along the direction of impulse or force).  But how could an object "decide" what to do in space, where is not resistance? How do I find the right size of filters for a lens? I was looking to get a circular polarizing filter and an UV filter for my Canon 550D, 18-135mm lens. When I checked online I found these filters came in different sizes, like 62mm, 67mm, 72mm, 77mm. How do I know which is right?  Also, will there be any degradation of photo quality if I put both these filters on the lens? Effect on loading a DC motor - load torque Could someone give a (detailed) explanation (perhaps with formulas) of how the load (torque) affects the armature current of a DC motor (separately excited or shunt or series)?  As the load torque increases the speed of the motor decreases, and assuming the terminal voltage stays constant, the EMF will become lower and thus the armature current will increase.  But could someone give me the formula that shows the relation between load torque and armature current?  Is it something like:   \$T_\text{developed} = T_\text{shaft} + T_\text{friction windage} + T_\text{load}\$ ,   where \$T_\text{developed} = K \, \text{flux}_\text{pole} \,I_\text{armature}\$ ?  But then you have  \$P_d = P_s + P_{f_w} + P_l   \$  where \$P_d = K \,\text{flux}_\text{pole}\, I_\text{armature}  \,\omega_m\$  In the last formula you can't see that as the increase due to \$P_\text{load }\$ the armature current increases because maybe someone can say the speed \$\omega_m\$ increases.  And how come at no-load the armature current and thus the torque developed is zero? Is it really zero, or do I have to assume there is some current flowing?  Could also explain me the principle of torque developed. Because at no load, the shaft torque is equal to torque developed, right? How do I get the current location of an iframe? How prove that $\lim\limits_{x\to+\infty}f(x)=\lim\limits_{x\to+\infty}f'(x)=0$ if $\lim\limits_{x\to+\infty}([f'(x)]^2+f^3(x))=0$? Question:     Let $f$ be differentiable on $[0,+\infty)$, such as$$\lim_{x\to+\infty}\left([f'(x)]^2+f^3(x)\right)=0$$show that $$\lim_{x\to\infty}f(x)=\lim_{x\to\infty}f'(x)=0$$   I think this problem is similar to  and (the  background)  For my problem I guess that if  $$\lim_{x\to+\infty}\left([f'(x)]^{2m}+f^{2n+1}(x)\right)=0,m,n\in N^{+},n,m&gt;1$$ then we have $\lim_{x\to\infty}f(x)=\lim_{x\to\infty}f'(x)=0$. How to yield from parallel tasks in .NET 4.5 How to send emails and avoid them being classified as spam? This is a  about how to handle email sent from your server being misclassified as spam.  For additional information you may find these similar questions helpful:                    Sometimes I want to send newsletters to my customers. The problem is, that some of the emails get caught as spam messages. Mostly by Outlook at the client (even in my own Outlook 2007).  Now I want to know what should be done to create "good" emails. I know about reverse lookup etc., but (for example), what about a unsubscribe link with an unique ID?  Does that increase a spam rating? I worked on a Word document and also saved the file on my flash drive.  When it asked me to save, I pressed the Replace document option by mistake.    I can see the missing work under the Properties menu. Will I be able to recover the lost work? Is there another saying/idiom similar to crocodile tears? Why can Arwen decide her mortality? Is gnu coreutils sort broken? Statistics interview questions Area of intersection between two circles Suppose you have 2 circles that intersect each other in such a way that each circle passes through the other's center. What is the area between the circle(or common area) i.e. area between the centres of the circles? Which packages should be loaded after hyperref instead of before? I installed Oneiric as a guest OS in VirtualBox using the Mini CD. I installed the ubuntu-desktop package and everything seems to work fine - well, except for a font issue with the terminal:    As you can see, some of the letters overlap and it's difficult to read. I checked the settings and it the checkbox "Use the system fixed width font" is checked with "Monospace | 12" selected underneath it.    Edit: running the commands mentioned in the comments:  $ fc-match "Monospace" DejaVuSansMono.ttf: "DejaVu Sans Mono" "Book" $ gconftool --get /desktop/gnome/interface/monospace_font_name Monospace 10 $ debsums ttf-dejavu-core /usr/share/doc/ttf-dejavu-core/AUTHORS                                        OK /usr/share/doc/ttf-dejavu-core/BUGS                                           OK /usr/share/doc/ttf-dejavu-core/NEWS.gz                                        OK /usr/share/doc/ttf-dejavu-core/README                                         OK /usr/share/doc/ttf-dejavu-core/changelog.Debian.gz                            OK /usr/share/doc/ttf-dejavu-core/copyright                                      OK /usr/share/doc/ttf-dejavu-core/langcover.txt.gz                               OK /usr/share/doc/ttf-dejavu-core/status.txt.gz                                  OK /usr/share/doc/ttf-dejavu-core/unicover.txt.gz                                OK /usr/share/fonts/truetype/ttf-dejavu/DejaVuSans-Bold.ttf                      OK /usr/share/fonts/truetype/ttf-dejavu/DejaVuSans.ttf                           OK /usr/share/fonts/truetype/ttf-dejavu/DejaVuSansMono-Bold.ttf                  OK /usr/share/fonts/truetype/ttf-dejavu/DejaVuSansMono.ttf                       OK /usr/share/fonts/truetype/ttf-dejavu/DejaVuSerif-Bold.ttf                     OK /usr/share/fonts/truetype/ttf-dejavu/DejaVuSerif.ttf                          OK Batch processing LiDAR files (.las to raster) with PyQGIS? for layers in glob.glob("*.las"):         processing.runalg("grass7:r.in.lidar", "input_las",5,1,"",1,100,"","",5,"","",False,True,True,"331289.57265,337990.42735,4558790.0,4560890.0",0,layers+"E:/las_new/.tif")      {'output': 'PNOA_2016_CAT_huso31_346-4560.lasE:/las_new/.tif'}   {'output': 'PNOA_2016_CAT_huso31_346-4562.lasE:/las_new/.tif'}   {'output': 'PNOA_2016_CAT_huso31_346-4564.lasE:/las_new/.tif'}   {'output': 'PNOA_2016_CAT_huso31_346-4566.lasE:/las_new/.tif'}   {'output': 'PNOA_2016_CAT_huso31_346-4568.lasE:/las_new/.tif'}   {'output': 'PNOA_2016_CAT_huso31_346-4570.lasE:/las_new/.tif'}   output_ras=("E:/las_Ras")   for layers in glob.glob("*.las"):     processing.runalg("grass7:r.in.lidar", "input_las",5,1,"",1,100,"","",5,"","",False,True,True,"331289.57265,337990.42735,4558790.0,4560890.0",0,layers+"output_ras.tif")      {'output': 'PNOA_2016_CAT_huso31_346-4560.lasoutput_ras.tif'}   {'output': 'PNOA_2016_CAT_huso31_346-4562.lasoutput_ras.tif'}   {'output': 'PNOA_2016_CAT_huso31_346-4564.lasoutput_ras.tif'}   {'output': 'PNOA_2016_CAT_huso31_346-4566.lasoutput_ras.tif'}   {'output': 'PNOA_2016_CAT_huso31_346-4568.lasoutput_ras.tif'}   {'output': 'PNOA_2016_CAT_huso31_346-4570.lasoutput_ras.tif'}     for layers in glob.glob("*.las"):     processing.runalg("grass7:r.in.lidar", "input_las",5,1,"",1,100,"","",5,"","",False,True,True,"331289.57265,337990.42735,4558790.0,4560890.0",0,"output_ras.tif")      {'output': 'output_ras.tif'}   {'output': 'output_ras.tif'}   {'output': 'output_ras.tif'}   {'output': 'output_ras.tif'}   {'output': 'output_ras.tif'}   {'output': 'output_ras.tif'}   I would like to batch process many LAS files to rasters using the PyQGIS algorithm. As you can see in the code, I think the final problem is with the output. It does not write the files out to my defined output pathway as tif files. Also how can I make sure the naming of output rasters are according to the original LAS they are generated from? What am I doing wrong here? Not a pro at Python. Started 3 days ago. White cloudy areas on bottom of new stainless steel skillet I just bought my first few pot and pans "Calaphon" stainless steel trip-ply  I used my new 10" skillet tonight, and cooked pork chops. I heated the pan first, added olive oil, let it heat, then added chops. After I was done, cleaned it immediately under water, and noticed white cloud looking areas on the bottom of the skillet. They won’t wash or rub off  Do I need to return these? I did buy them for the look as well.