Show, if $f:X-&gt;\mathbb R$ is continuous at point $a \in X$, then it is bounded in a neighborhood of the point $a$. That is, it exists $\delta&gt;0$ such that $U_{\delta}=X \cap (a-\delta,a+\delta)$ or $f(U_{\delta})$ is bounded. Help me, please
I have this question :  Proof if $f$ continuous in $x_0$ then there is a neighbourhood of $x_0$ so $f$ bounded.  I want to know if my proof is valid :  If continuous in $x_0$ then :  $$\lim_{x \to x_0} f(x)= f(x_0)$$  Therefore :  All $\epsilon&gt;0$ exists $\delta&gt;0$ so any $x$ that implies $|x-x_0|&lt;\delta$ implies $|f(x)-f(x_0)|&lt;\epsilon$.  Therefore for $\epsilon=1$ there is $\delta&gt;0$ so any $x$ that implies $|x-x_0|&lt;\delta$ implies $|f(x)-f(x_0)|&lt;1$.  $|x-x_0|&lt;\delta \rightarrow x_0-\delta&lt;x&lt;x_0+\delta$.  Lets choose $x_1,x_2$ that implies $x_0-\delta&lt;x_1&lt;x_2&lt;x_0+\delta$  Therefore in $[x_1,x_2]$ from Weierstrass from theorem bounded there.
How to insert text into a root-owned file using sudo? What effect does the Barr body have, in relation to female Turner syndrome? Why do persons with  have  developmental abnormalities, when normal XX-females do not, even though they only have 1 active X chromosome? From what I know, one X-chromosome is inactivated into a Barr body in normal XX-females, but these obviously don't show the symptoms of Turner syndrome. What function does the inactive chromosome have? Is it safe to set xp_cmdshell value 1? I need to use xp_cmdshell in one of SQL job.So is it safe to reconfigure xp_cmdshell to 1 or can i set value 1 in the starting of SQL statements and set it back to 0 in the end of SQL Statement ? Why does everyone's wand still work for them? How do I quickly switch between workspaces in unity? Encouraging software publishers to use StackExchange as support mechanism Whenever I deal with an application publisher's custom support mechanism because I can't see any evidence of activity on StackExchange, I always wish they'd just migrate to StackExchange, particularly when I get a prompt response from them. What a waste, I think, that my question and answer weren't captured on SE.  In cases where there is some activity for the application on SE, I'll post my question there and, if I get no response, reference the SE question in my email follow-up. In the case of , that had a happy ending where the support person posted the answer to the SE site at the same time they emailed me.  My related meta questions are:   Are there any obvious downsides to publishers effectively "giving up" their custom support mechanisms, or at least their custom forum, in favor of the appropriate SE site? Are there any obvious downsides to SE for publishers making this transition? Is SE staff engaged in any systematic marketing efforts to get publishers to make this transition? Is there any SE documentation which lists the advantages of publishers making this transition? Are the known exoplanets representative of the population of exoplanets? An  states that most of the 715 newly announced exoplanets orbit very close to their host stars.   Do the theories predict that this is a representative sample and that most planets are very close to their host stars? Or is it just that our current detection techniques are predisposed to finding only those planets close to their stars (because the sensors look for dimming of the host star for example)? machine wakes only on local magic packet I've got a Debian server, which I would like to wake up remotely using a tool like wakeonlan.  I have assigned the server a static internal IP 192.168.0.14 and set up the appropriate port forwarding in the router (which otherwise has a very limited admin interface).  When I am logged in to the same network, I can wake the server using the subnet IP, but I cannot wake the server using the public IP of the router.  What I do not understand: When the server is running and I listen on the server with netcat nc -ulv -p 9, I can see the packet coming in in both cases (both when sending to the subnet IP and when sending to the public IP of the router).  If the packet arrives in both cases, why does the machine wake up only in one of them? How to compile bibtex with TeXstudio? I am trying to use bibtex for the first time. I am on windows 7, using TeXstudio. I am trying to follow the steps of .  I have created a .bib file, put it in the right directory and written the code in my .tex file. However, I do not know how to compile bibtex with TeXstudio (I looked around in the menus but didn't find it).   Most tutorials I found don't explain how to do it, or say to run a command, which I'm not sure how to do.  Any help much appreciated! How to enable/disable a specific graphic card? How to bind 'touchstart' and 'click' events but not respond to both? If the center of a group $G$ is of index $n$, prove that every conjugacy class has at most $n$ elements. (This question is from Dummit and Foote, page 130, 3rd edition.)  Here is my attempt: we have  $|G| =|C_G (g_i)| | G : C_G (g_i)| $  $|C_G (g_i)| =|Z(G)| \cdot  |C_G (g_i):Z(G)| $.  Then   $|G|= |Z(G)|\cdot |C_G (g_i):Z(G)|\cdot | G : C_G (g_i)| $.  Then  $n = |C_G (g_i):Z(G)|\cdot  | G : C_G (g_i)| $.  But $|C_G (g_i):Z(G)|$ is positive integar as  $ Z(G)$ is   subgroup of  $C_G (g_i)$.  If $| G : C_G (g_i)|$ bigger than $n$, then $|C_G (g_i):Z(G)|\cdot | G : C_G (g_i)|$ is bigger than $n $.  But   $|C_G (g_i):Z(G)|\cdot | G : C_G (g_i)|=n$, contradiction.  Then $| G : C_G (g_i)|\le n$ and this completes the proof.   Is this proof right or not ??!!! Solve $p_4(x) = x^4 −(2m + 4)x^2 + (m−2)^2 $such that $p_4$ is a product of two non-constant integer-coeficient polynomials I'm having trouble getting the starting idea for a problem I've been presented with:  I need to find values for m (integer) such that the following polynomial     $p_4(x) = x^4 −(2m + 4)x^2 + (m−2)^2$; $m$ is integer   can be described as the product of two non-constant integer-coeficient polynomials.  Any help/pointers/tips on how to go about this are greatly appreciated!  Edit: I meant polynomials with integer coeficients, but for some reason typed positive. How to fix a stripped pedal crank The threads on my bike's pedal crank just became fully stripped and the pedal fell off, though the pedal itself is fine. Is there any way I can fix this without fully replacing the pedal crank? Since I first learned English, I have been holding this understanding that "and", as a conj. but unlike "but",  can only connect two clauses, not two sentences ended with periods.  But recently, I have seen so many prints, either in entertainment or in academia, where "And" is popularly used in the beginning of a sentence. It seems like the author is trying to connect the sentence just right before and the sentence following "And" in some intended meaning which I don't quite get.  I was wondering if I have been wrong all the time, or if there is a new trend that I fail to understand and accept? How to understand such usage of "and" in rigorous English grammar? If I am right, why it is used differently from "but" in terms of what I mentioned at the beginning of this post? Integration of secant $$\begin{align} \int \sec x \, dx  &amp;= \int \cos x \left(  \frac{1}{\cos^2x} \right)  \, dx \\ &amp;= \int \cos x \left(  \frac{1}{1-\sin^2x} \right) \, dx \\ &amp; = \int\cos x\cdot\frac{1}{1-\frac{1-\cos2x}{2}} \, dx \\ &amp;= \int \cos x \cdot\frac{2}{1+\cos2x} \, dx  \end{align}$$  I am stuck in here. Any help to integrate secant? Before I start, don't lecture me on the fact that FAT32 is old and hasn't been updated for nearly 20 years. Right now I have sorted that. I am trying to install Ubuntu Linux onto a 64GB flash drive and it wants me to format it to one of the EXT file systems.  The problem is if I put the flash drive into a Windows PC, I can't use it. I also can't install the required software onto every Windows PC that I will use. When I tried to install it on FAT32 I got an error and there is no option for NTFS.  Does anyone know how I could do this? Maybe a Windows based tool? Thanks in advance. How do  form? Why are they called the "time-reversed" versions of a black hole? What's an "eternal" black hole? How come a white hole is only encountered if you remove the star from the space-time surrounding the black hole? Why is it that if you add any matter to the area that the white hole disappears? Any explanation greatly appreciated. Why are two permutations conjugate iff they have the same cycle structure? I am not experienced in Number Theory but what I know is that some results of this filed are applicable in other areas, e.g. algebra. For sure FLT made (and makes) people be interested in Number Theory leading to the development of new methods which can be themselves applied not only for the proof of FLT (like financial problems motivated somehow development of stochastic analysis). I am interested - if there are applications or implications of FLT itself?  More precisely: if the fact "for each $n\geq3$ there are no integer solutions of $a^n+b^n=c^n$" leads to solutions of problems which are not in the field of Number Theory?   I would specify that I wonder about some problems which are already formulated: since FLT is known for more then 300 years I am pretty sure that there were formulated hypothesis which follow from FLT directly (if there are such hypothesis not in the field of Number Theory).