I don't know what the answer is but I tried to get some information from the hypothesis and have reached till the result that $f$ is an homeomorphism.  This is because $f$ is bijective and continuous as well as it's inverse is also continuous because $f \circ f=f^{-1}$ and composition of continuous functions is continuous.  What should be my next step? or this is not the way to reach the answer?
Let $f:\mathbb R \to \mathbb R$ be a continuous function such that $f(f(f(x)))=x,\forall x \in \mathbb R$ , then is it true that $f(x)=x , \forall x \in \mathbb R$ ? I can see that $f$ is bijective , hence strictly monotone . But I cannot progress further . Please help. Thanks in advance
Mac turns off randomly and has lines on screen For some reason my iMac keeps turning off around 5 to 10 minutes into boot.  Also just recently it turns on but has shaded lines going down the screen. I have not installed any weird programs and i have not used it in around 3 months.   Any ideas on what the issue is, i don't have any other information about the issue besides whats happening when i use it.  The image i'm showing you was found on google but this looks exactly like mine does.  The model of my imac is the late 2011 model. If a factor variable (e.g. gender with levels M and F) is used in the glm formula, dummy variable(s) are created, and can be found in the glm model summary along with their associated coefficients (e.g. genderM)  If, instead of relying on R to split up the factor in this way, the factor is encoded in a series of numeric 0/1 variables (e.g. genderM (1 for M, 0 for F), genderF (1 for F, 0 for M) and these variables are then used as numeric variables in the glm formula, would the coefficient result be any different?  Basically the question is: does R use a different coefficient calculation when working with factor variables versus numeric variables?  Follow-up question (possibly answered by the above): besides just the efficiency of letting R create dummy variables, is there any problem with re-coding factors as a series of numeric 0,1 variables and using those in the model instead? Example of the equality of an inequality This question is related to Daniel Fischer's answer . Suppose $f$ is a real $C^{1}$ function on $[0, 1]$ such that $f(0) = 0$ and $\int_{0}^{1}f'(x)^{2}\, dx \leq 1$. Then (essentially by Cauchy-Schwarz), we have $\left|\int_{0}^{1}f(x)\, dx\right| \leq 2/3$ as Daniel Fischer stated in his answer.  My question is: Is there an example of a function such that we have equality in $\left|\int_{0}^{1}f(x)\, dx\right| \leq 2/3$? I was thinking of maybe a smoothed version of a function which takes the value $0$ on $[0, 1/2)$ and $4/3$ on $(1/2, 1]$ but that seems complicated to construct, moreover, the derivative in some neighbourhood of $1/2$ might be hard to control. If $x>0$, $\,x^{1/n}$ tends to $1$ as $n\to\infty$ I am looking for a proof possibly using the sandwich theorem and/or Bernoulli's inequality for proving the following statement:  If $x&gt;0$ then $x^{1/n}$ tends to $1$. A classic exponential inequality: $x^y+y^x>1$ This is a classic problem, but I haven't seen it on the site before.     Suppose $x$ and $y$ are real numbers that both lie in the interval   $(0,1)$. Prove  $$x^y+y^x&gt;1.$$   The following spoiler box contains a hint that indicates the solution I know. Does anyone know a more natural solution?       Use Bernoulli's inequality to show $$x^y\ge \frac{x}{x+y}.$$   Edit: I recently learned the source is the 1996 French math olympiad. Firefox 37.0.2 not opening (removing firefox did not help) runing ubuntu 14.04 LTS SSH failed to start - Missing privilege separation directory: /var/run/sshd Force.com IDE 35.0.0.201510211056 in Eclipse missing run test class option Force.com IDE 35.0.0.201510211056 and Force.com IDE Debugger 35.0.0.201510211056 in eclipse but it doesn't show this option to run test class, is their any way i can see it because it show the Apex_Test_Result panel but running a test class option is missing. plz help Selecting all files in directory for merging in QGIS Processing? I'm writing a generic script which involves writing shapefiles to a directory which are then merged together. After writing the files out to the Output folder, I'm trying to get the saga:mergeshapeslayers algorithm to merge all the files in the Output folder. I have used the Model Builder and although it is helpful to an extent, I find that it is used for specific purposes whereas I am attempting to make a script for generic purposes.   Code:  ##Test=name ##Select_folder=folder ##Result=output vector  import os import glob  path_1 = Select_folder path = path_1 os.chdir(path)  def function():      output = glob.glob(path_1 + './*.shp')     x = 0      while output[x]:         for fname in glob.glob("*.shp"):             outputs_1 = processing.runandload("qgis:fieldcalculator", output[x], 'Number', 1, 10, 0, True, 1 , "C:\Users\Me\Desktop\Output\\"  + fname)             multiple_0 = glob.glob("*.shp")             x = x + 1          if x + 1 &gt; len(output):             processing.runalg("saga:mergeshapeslayers", output[0], ";".join(multiple_0) , Result)             break         else:             continue  if path_1:     function() else:    pass It's not like there's anything else to do.   Shouldn't the speaker say: "There's something else to do." ? so what's the meaning of the sentence and what's the difference between the two of them? Thank you, Ubuntu 14.04 slow wireless connection This is a follow-up to .  Suppose you want to denote vectors by upright bold letters; then \mathbf{u} works fine, except that if it's a Greek letter you have to use \boldsymbol{\omega} instead.  Question: Is it possible to define a generic vector command in LaTeX which automatically chooses between \mathbf and \boldsymbol, so that one can write just \vect{u} and \vect{\omega} without having to treat the Greek symbols specially by hand?  (Just to be clear: \newcommand{\vect}[1]{\boldsymbol{#1}} is no good in this situation, since \boldsymbol{u} gives bold italic instead of roman.) Switches needed in an n-location multiway switching system we can construct a two location multiway switching system with two 3-way switches:    They also teach us how to perform three location multiway switching with two 3-way switches and one 4-way switch:    Is it possible to make a similar system (three locations, all able to switch the light in any situation) with only three 3-way switches?   What can be said in general about the switches needed to make an n-location multiway switching system? Identify cartoon about a set of fairy/demon twins in a far future where the demon uses propaganda to take over the world I'm looking for a cartoon from the 70's (I believe) which is about a set of twins; one plump "fairy" and a skeletal "demon". The "fairy" has magic powers and ends up living a life of pleasure, while the "demon" flees and turns evil.  When the "demon" finds his new weapon in the ruins of the ancient world (roughly modern day) he plots to enslave the world; the weapon in question is a set of propaganda movies with Adolf Hitler. Seeking open source solution to access rasters in File Geodatabase? I'm looking for a way to access raster layers in an Esri File-GeoDatabase without using ArcObjects or ArcPy, specifically I would like to access the raster layers with GDAL. I understand that the GDAL FileGDB and OpenFileGDB give access to vector datasets within File-GeoDatabases, and that ESRI's own FileGDB C++ API does not support raster access.   Is anyone out there aware of any open source solutions? What is the main difference between StringBuffer and StringBuilder? Is there any performance issues when deciding on any one of these? How come unpolarized light does not undergo destructive interference? Imagine a battery with the positive terminal connected to ground.  The positive side has a high eletrical potential, ground obviously has a lower electric potential.  Electricity wants to seek a lower potential, and there's no significant resistance between the terminal and the ground.  Shouldn't electricity flow?  I know it doesn't but I don't understand why.  I know you have to close the loop, but why is this necessary? Why doesn't the tilde (~) expand inside double quotes? According to  and my own understanding, the tilde expands to the home directory:  $ echo ~ /home/braiam   Now, whenever I want the shell expansion to work, i. e. using variable names such $FOO, and do not break due unexpected characters, such spaces, etc. one should use double quotes ":  $ FOO="some string with spaces" $ BAR="echo $FOO" $ echo $BAR echo some string with spaces   Why doesn't this expansion works with the tilde?  $ echo ~/some/path /home/braiam/some/path $ echo "~/some/path" ~/some/path I had an little issue when loggin in with Safari on Mac OS X (El Capitan).   When entering the password in Safari, there is a key symbol that lets you choose among any password that is stored in your keychain. This button interferes with the "forgot?" button that Stack Exchange adds and you can't press the Safari button.     Please move this "forgot?" button below the text field, like most other sites implement this.