I need help with this question as I'm not sure Let $f$ be a continuous function on $K$ which is closed in the real line. Show that there exists a continuous $F$ on all of $R$ such that $F(x)=f(x)$ on $K$.  So I'm not sure which theorem to use in Folland to prove this. If anyone can hint on what to do, I'd be grateful.
Any help with this problem is appreciated. It comes up in the context of Lusin's theorem, where it was assumed to be true.     Suppose $E$ is a closed subset of $\mathbb{R}$ and $f:E\rightarrow \mathbb{R}$ is continuous. There exists a continuous function $g: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(x)=g(x)$ for $x \in E$ and $g$ satisfies $$\sup_{x \in \mathbb{R}} |g(x)| \leq \sup_{x \in E} |f(x)|\;.$$
What are the differences between executing shell scripts using "source file.sh", "./file.sh", "sh file.sh", ". ./file.sh"? I cannot mount Windows 8 partition disk on Ubuntu 12.10.  My Windows 8 is installed on "C" partition, and I have another "D" partition.  I can easily mount "D" partition, see files and make changes on them. Actually I also have a problem with this: when I copy a file from ubuntu to "D", I cannot see this file on Windows 8.  My main problem is when I try to mount "C" partition I get this:     Unable to mount 121 GB Volume   Error mounting /dev/sda3 at /media/furkan/06DEC78FDEC77589: Command-line `mount -t "ntfs" -o "uhelper=udisks2,nodev,nosuid,uid=1000,gid=1000,dmask=0077,fmask=0177" "/dev/sda3" "/media/furkan/06DEC78FDEC77589"' exited with non-zero exit status 14: Windows is hibernated, refused to mount.   Failed to mount '/dev/sda3': Operation not permitted   The NTFS partition is hibernated. Please resume and shutdown Windows   properly, or mount the volume read-only with the 'ro' mount option, or   mount the volume read-write with the 'remove_hiberfile' mount option.   For example type on the command line:           mount -t ntfs-3g -o remove_hiberfile /dev/sda3 /media/furkan/06DEC78FDEC77589 Introduction to string theory I am in the last year of MSc. and would like to read string theory. I have the Zwiebach Book, but along with it what other advanced book can be followed, which can be a complimentary to Zwiebach. I would like a more mathematically rigorous book or lecture notes along with Zwiebach.   Specifically, mention whether the book discusses string theory   Rigorously? Intuitively?   What's the scope of the book? Does it cover the advanced materials, e.g. Matrix string theory, F-theory, string field theory, etc. Maybe even String Phenomenology? Here's my code:  x = 1.0 y = 100000.0     print x/y   My quotient displays as 1.00000e-05.  Is there any way to suppress scientific notation and make it display as 0.00001? I'm going to use the result as a string. Adding layers to GeoPackage using PyQGIS C Scanf suddenly stopped reading in values Show all of my question/answers to me even if they are deleted How do I remove code duplication between similar const and non-const member functions? A friend of mine accidentally left a carton of eggs on her counter, unrefrigerated, for three days. The eggs had been previously refrigerated both at the store and at home. Now she's planning to do some more cooking which requires eggs, and is wondering if it's still safe to use them for baking. I believe she is planning on baking cookies with them, so they would be baked at fairly high temperatures for probably at least 10 minutes. Would this be safe, or are eggs left unrefrigerated for that long not safe for consumption? Installing "firmware-b43-installer" offline What really is an indeterminate form? We can apply l’Hôpital’s Rule to the indeterminate quotients $ \dfrac{0}{0} $ and $ \dfrac{\infty}{\infty} $, but why can’t we directly apply it to the indeterminate difference $ \infty - \infty $ or to the indeterminate product $ 0 \cdot \infty $?  Furthermore, why can’t we call $ \infty + \infty $ and $ \infty \cdot \infty $ indeterminate forms?  I’m new to calculus, so please clear up my concepts if you can. Thanks! When to use static methods Using C# and WPF under .NET (rather than  or console), what is the correct way to create an application that can only be run as a single instance?  I know it has something to do with some mythical thing called a mutex, rarely can I find someone that bothers to stop and explain what one of these are.  The code needs to also inform the already-running instance that the user tried to start a second one, and maybe also pass any command-line arguments if any existed. Why doesn't **sort** sort the same on every machine? Using the same sort command with the same input produces different results on different machines.  How do I fix that? I was trying to change my username and also my home directory (/home/username) and my system started to crash. I deleted the passwd file but I had a backup named passwd_bkp. I tried to rename this passwd_bkp as passwd and it did not work. No commands were being executed... I was in a terminal window.   I restarted my system and now I cannot log in. GRUB gives two options: Linux and recovery mode.   I tried to open a sessions as root but it says the filesystem is corrupted. I cannot access my files.  Did I lose all my files? Suppose we have a set of primes $p_1,\dots,p_t$. Prove that  $\log p_1,\dots,\log p_t$ is linear independent over $\mathbb{Q}$. Now, this implies $ \sum_{j=1}^{t}x_j\log(p_j)=0 \iff x_1=\dots=x_t=0$.  I think I have to use that fact that every $q\in\mathbb{Q}$ can be written as $\prod_{\mathcal{P}}$, where $n_p$ is a unique sequence ($n_2$,$n_3$,$\dots$) with domain $\mathbb{Z}$. Here, $\mathcal{P}$ denotes the set of all integers.  Now how can I use this to prove the linear independency? How can I disable the “News and Interests” Widget on the Windows 10 Taskbar? On Windows 10 version 1909 and newer releases, Microsoft has added a &quot;News and Interests&quot; widget on the Taskbar (see ):  How can I remove this Widget from the Taskbar? Intellisense in IntelliJ no longer working JavaScript for...in vs for Do you think there is a big difference in for...in and for loops? What kind of "for" do you prefer to use and why?  Let's say we have an array of associative arrays:  var myArray = [{'key': 'value'}, {'key': 'value1'}];   So we can iterate:  for (var i = 0; i &lt; myArray.length; i++)   And:  for (var i in myArray)   I don't see a big difference. Are there any performance issues? How prove this inequality $\frac{a}{b+3}+\frac{b}{c+3}+\frac{c}{d+3}+\frac{d}{a+3}\le 1$ Question:  let $a,b,c,d\ge 0$,such $$a^2+b^2+c^2+d^2=4$$ show that $$\dfrac{a}{b+3}+\dfrac{b}{c+3}+\dfrac{c}{d+3}+\dfrac{d}{a+3}\le 1$$  My try: By Cauchy-Schwarz inequality,we have $$\sum_{cyc}\dfrac{a}{b+3}\le\sqrt{(\sum_{cyc}a^2)\left(\sum_{cyc}\dfrac{1}{(b+3)^2}\right)}$$ then we have only prove this $$\sum_{cyc}\dfrac{1}{(a+3)^2}\le \dfrac{1}{4}?$$ This is  not true,in fact, we have $$\sum_{cyc}\dfrac{1}{(a+3)^2}\ge \dfrac{1}{4}?$$ because we have $$\dfrac{1}{(a+3)^2}\ge\dfrac{5-a^2}{64}$$ this is true because $$\Longleftrightarrow \dfrac{(a-1)^2(a^2+8a+19)}{64(a+3)^2}\ge 0$$ so $$\sum_{cyc}\dfrac{1}{(a+3)^2}\ge\sum_{cyc}\dfrac{5-a^2}{64}=\dfrac{1}{4}$$ can see:  This methods is from: can  see:  By the way I have see this three variable inequality  let $a,b,c$ be non-negative numbers such that $$a^2+b^2+c^2=3$$  show that $$\dfrac{a}{b+2}+\dfrac{b}{c+2}+\dfrac{c}{a+2}\le 1$$  proof: By expanding,the inequality becomes $$ab^2+bc^2+ca^2\le abc+2$$ without loss of generality,assume that  $$\min(a,b,c)\le b\max(a,b,c)$$ then \begin{align*} 2-ab^2-bc^2-ca^2+abc&amp;=2-ab^2-b(3-a^2-b^2)-ca^2+abc\\ &amp;=(b^3-3b+2)-a(b^2-ab+ca-bc)\\ &amp;=(b-1)^2(b+2)-a(b-a)(b-c)\ge 0 \end{align*} Equality occurs for $(a,b,c)=(1,1,1)$ and also for $(a,b,c)=(0,1,\sqrt{2})$ or any cyclic permutation.  so my Four-inequality  variable inequality,can use this methods $$\dfrac{a}{b+3}+\dfrac{b}{c+3}+\dfrac{c}{d+3}+\dfrac{d}{a+3}\le 1$$ $$\Longleftrightarrow a^2cd+3a^2c+3a^2d+9a^2+ab^2d+3ab^2+abc^2-abcd+3ac^2+9ac+3b^2d+9b^2+3bc^2+bcd^2+3bd^2+9bd+9c^2+3cd^2+9d^2-81\le 0$$ $$\Longleftrightarrow a^2cd+3a^2c+3a^2d+ab^2d+3ab^2+abc^2+3ac^2+9ac+3b^2d+3bc^2+bcd^2+3bd^2+9bd+3cd^2\le 45$$ $$\Longleftrightarrow a^2(cd+3c+3d)+ab^2d+3ab^2+abc^2+3ac^2+9ac+3b^2d+3bc^2+bcd^2+3bd^2+9bd+3cd^2\le 45$$ $$\Longleftrightarrow (4-b^2-c^2-d^2)(cd+3c+3d)+ab^2d+3ab^2+abc^2+3ac^2+9ac+3b^2d+3bc^2+bcd^2+3bd^2+9bd+3cd^2\le 45$$  Then  I can't  Thank you  very  much