Lat $a&lt;b$ I have to Show that there exist smooth functions $\psi:\mathbb{R}\rightarrow\mathbb{R}$ such that  $\psi(x) = \begin{cases} &gt;0 &amp;\mbox{if } a&lt;x&lt;b \\   =0 &amp; \mbox{if }  x\leq a \text{ or } x\geq b. \end{cases}$  I also have the result   that the function   $f(x) =\begin{cases} e^{-\frac{1}{x}}&amp; \mbox{ } x&gt;0 \\ 0, &amp;\mbox{ } x\leq 0 \end{cases}$  is smooth and that the every derivative in $0$ vanishes.  For the left side I can just change the function $f(x)$  $f_a(x)=\begin{cases} e^{-\frac{1}{x-a}}&amp; \mbox{ } x&gt;a \\ 0, &amp;\mbox{ } x\leq a \end{cases}$  Because I already know that $\lim_{x\downarrow 0}f(x)=0$ then also $ \lim_{x\downarrow a}f_a(x)=0$.  For the right side I want that the function has the same behaviour but I don't know how I can construct such a function.   Hints are appreciated
Is it possible to find a smooth function $p$ such that $$p(x)=0 \text{ if } x\le -1$$ $$p(x)=0 \text{ if } x&gt; 0$$ $$0\le p(x) \le 1 \text{ if } -1&lt;x&lt;1$$ $$p(x) = 1 \text{ if } -\frac{1}{2}\le x \le \frac{1}{2}$$ Hint: The following function has continuous derivatives of all orders $$f(x) = 0 \text{ if } x\le 0$$ $$f(x) = e^{-\frac{1}{x^2}} \text{ if } x&gt;0$$  My thought: Considering the left-half of the function $p$, I tried something like $e^{-\frac{1}{\tan^2(x\pi)}}$. The shape looks good but I am not sure about the smoothness at $-1$ and $-\frac{1}{2}$. For $x=-1$, since $\tan$ is infinitely differentiable so we have smoothness there. But for $x=-\frac{1}{2}$, I am not sure about the smoothness after &quot;gluing&quot; with $y=1$.
I noticed that the locked message still references the FAQ. See :  locked by  yesterday This question exists because it has historical significance, but it is not considered a good, on-topic question for this site, so please do not use it as evidence that you can ask similar questions here. This question and its answers are frozen and cannot be changed. More info: .  Since the FAQ has been replaced by the , this message should provide a link there instead. Currently, the link actually takes you to the ; even if it doesn't link to the help center, the text should probably be something like &quot;More info: &quot;. Proving $\mathbb{Q}$ isn't homeomorphic to $\mathbb{Z}$ Prove that $\mathbb{Q}$ isn't homeomorphic to $\mathbb{Z}$  I know that a homeomorphism is a bijection function from two sets where both the function and its inverse are continuos.   So I figure that if I show that there is no continuos invective mapping from $\mathbb{Q}\rightarrow\mathbb{Z}$ then I have done the proof. But I do not know how to do this (if his is the right way). How can I start to show this? How do I unit test (using xUnit) a class that has internal private methods, fields or nested classes? Or a function that is made private by having  (static in C/C++) or is in a private () namespace?  It seems bad to change the access modifier for a method or function just to be able to run a test. A calligraphic font with small (lowercase) letters Is there a widely available calligraphic font that (1) contains small (lowercase) letters and (2) has a recognizably calligraphic capital letter O?  Background. (1) In mathematics (algebraic geometry), , the latter being written in calligraphic script (standard notation for sheaves). However, most books end up typesetting the sheaf as follows    pulling the small letters from the italics of a completely different font.  (2) The calligraphic capital letter O plays a special role in algebraic geometry, too, and most authors insist on it being recognizably calligraphic (i.e. not just a closed round loop)    Edit. The following are achieved with extra kerning from @egreg's and @UlrikeFischer's answers (in chronological order)    @egreg's answer  \makeatletter \DeclareFontEncoding{LS1}{}{} \DeclareFontSubstitution{LS1}{stix}{m}{n} \DeclareMathAlphabet{\mathcal}{LS1}{stixscr}{m}{n} \makeatother  \DeclareMathOperator{\sHom}{\mathcal{H\mkern-7mu o\mkern-2.5mu m\mkern-1.5mu}} \DeclareMathOperator{\sExt}{\mathcal{E\mkern-4.5mu x\mkern-2.5mu t\mkern-1mu}} \DeclareMathOperator{\sEnd}{\mathcal{E\mkern-4mu n\mkern-4.5mu d\mkern-1mu}}       @UlrikeFischer's answer  \DeclareFontFamily{U}{mathc}{} \DeclareFontShape{U}{mathc}{m}{it}% {&lt;-&gt;s*[1.03] mathc10}{} \DeclareMathAlphabet{\mathcal}{U}{mathc}{m}{it}  \DeclareMathOperator{\sHom}{\mathcal{H\mkern-3mu om}} \DeclareMathOperator{\sExt}{\mathcal{E\mkern-3mu xt}} \DeclareMathOperator{\sEnd}{\mathcal{E\mkern-3mu nd}} dpkg --configure -a  Setting up linux-image-3.13.0-35-generic (3.13.0-35.62) ... Running depmod. update-initramfs: deferring update (hook will be called later) The link /initrd.img is a dangling linkto /boot/initrd.img-3.13.0-35-generic vmlinuz(/boot/vmlinuz-3.13.0-35-generic ) points to /boot/vmlinuz-3.13.0-35-generic  (/boot/vmlinuz-3.13.0-35-generic) -- doing nothing at /var/lib/dpkg/info/linux-image-3.13.0-35-generic.postinst line 491. Examining /etc/kernel/postinst.d. run-parts: executing /etc/kernel/postinst.d/apt-auto-removal 3.13.0-35-generic /boot/vmlinuz-3.13.0-35-generic run-parts: executing /etc/kernel/postinst.d/initramfs-tools 3.13.0-35-generic /boot/vmlinuz-3.13.0-35-generic update-initramfs: Generating /boot/initrd.img-3.13.0-35-generic  gzip: stdout: No space left on device E: mkinitramfs failure cpio 141 gzip 1 update-initramfs: failed for /boot/initrd.img-3.13.0-35-generic with 1. run-parts: /etc/kernel/postinst.d/initramfs-tools exited with return code 1 Failed to process /etc/kernel/postinst.d at /var/lib/dpkg/info/linux-image-3.13.0-35-generic.postinst line 1025. dpkg: error processing package linux-image-3.13.0-35-generic (--configure):  subprocess installed post-installation script returned error exit status 2 dpkg: dependency problems prevent configuration of linux-image-extra-3.13.0-35-generic:  linux-image-extra-3.13.0-35-generic depends on linux-image-3.13.0-35-generic; however:   Package linux-image-3.13.0-35-generic is not configured yet.  dpkg: error processing package linux-image-extra-3.13.0-35-generic (--configure):  dependency problems - leaving unconfigured dpkg: dependency problems prevent configuration of linux-image-generic:  linux-image-generic depends on linux-image-3.13.0-35-generic; however:   Package linux-image-3.13.0-35-generic is not configured yet.  linux-image-generic depends on linux-image-extra-3.13.0-35-generic; however:   Package linux-image-extra-3.13.0-35-generic is not configured yet.  dpkg: error processing package linux-image-generic (--configure):  dependency problems - leaving unconfigured Errors were encountered while processing:  linux-image-3.13.0-35-generic  linux-image-extra-3.13.0-35-generic  linux-image-generic Converting from DMS to DD using Python in Field Calculator? 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There are no "join" buttons in the game itself.  I've read somewhere that Spawn version is a trial version but can be used to join Multiplayer games. If they can, is it possible for them to play the entire game on LAN or is trial restricted from choosing some classes/accessing some areas?  Or are there some other ways to join? Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that $f(0) = 0$ and $f'(x) = [f(x)]^{2}$, $\forall x \in \mathbb{R}$. Show that $f(x) = 0$,  $\forall x \in \mathbb{R} $.  I first (unsuccessfully) tried using the Mean Value Theorem, but this is in the Integrals chapter so the solution probably involves them. Can't really see where integrals come in here though.  What I've got so far:   (i) Since $f$ is differentiable, thus it is continuous and, hence,  integrable. Therefore $f^2$ is also integrable and as $f'=f^2$, $f'$ is too.   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I asked a question, and then, someone answered it.    I clicked "Edit" to add a comment, and I would like to press Enter to go to a new line, but doing so submitted my comment and ended the edit instead. How to change sign text color in Minecraft? I have gone through every single website and video to try to make a sign with colored text, but nothing works. Can anyone help me to get a sign to say:  Place Block(dark blue and bold text)  I really want this for a map I am making and I need help. I was wondering about generic installations of all applications in linux. And what that means? Well, when I was using windows I knew that if I want to install an application I am double-clicking the .exe file and then next,next,next.  In linux, I have understood that maybe there is a common (not generic) way to install any application. Installing from source maybe? Well is there any step by step method that can be used to install  application like in windows or not?  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