Show that if $f : [a, b] \to \mathbb{R}$ is non-negative and continuous on $[a, b]$ and  $$\int_{a}^{b} f(x)dx = 0$$ then  $f(x)=0$, for all $x ∈ [a, b]$.  I'm having difficulties in proving things to do with integrals, how do I start this?
If $f$ is continuous, nonnegative on $[a, b]$, show that $\int_{a}^{b} f(x) dx = 0$ iff $f(x) = 0$  "$\Rightarrow$" Assume by contradiction that $f(x) \neq 0$ for some $x_0 \in [a, b]$. Without loss of generality, assume that $f(x_0) &gt; 0$. Since $f$ is continuous, $\exists \delta &gt; 0$ such that $f(x) &gt; 0,  \forall x \in (x_0 - \delta, x_0 + \delta)$. Since $f$ is continuous on $[a, b]$, then it is Riemann integrable. Therefore $\int_{a}^{b} f(x) dx = \sup L(f, p)$ for some partition $p$. Now what I want to do is to show that the lower sum, $L(f, p) &gt; 0$, in that case its supremum is $&gt;0$, and that contradicts the condition that $\int_{a}^{b} f(x) dx = 0$. However, I don't know how to articulate it.  "$\Leftarrow$" If $f(x) = 0, \forall x \in [a, b]$, then $\int_{a}^{b} f(x) dx = \int_{a}^{b} 0 dx = 0$
When I am in texture mode one of my objects looks like it is in solid mode. The model is textured like all other models in the scene, but it doesn't look like it is in texture mode.  Screenshot will explain better:      What this model looks like. What all models should look like in texture mode. Find out which application listens to specific key bindings? Is there an application to know which key is bound by which process and therefore not usable in another application? apt-get build-dep is unable to find a source package I have already followed  to disable middle mouse button paste on my Ubuntu 12.04.   Works like a charm.   Now I am trying to achieve the same on my Linux Mint 17. When I try to  sudo apt-get build-dep libgtk2.0-0   it gives me the following output:   Reading package lists... Done Building dependency tree        Reading state information... Done Picking 'gtk+2.0' as source package instead of 'libgtk2.0-0' E: Unable to find a source package for gtk+2.0   For me it looks like apt-get is somehow "resolving" 'libgtk2.0-0' to 'gtk+2.0', but then does not find any package named like that.   EDIT:  although I am now able to compile the program (see my answer), I still do not know what  Picking 'gtk+2.0' as source package instead of 'libgtk2.0-0' is supposed to mean. Any insight on this would be appreciated, thanks! How do I reboot a Ubuntu Server from the command line? sudo reboot now never seems to work for me. Because its a server, I don't have a window manager installed, so I can't do it from a GUI (which always seems to work).    I've had the problem for years on different versions of the operating system. It happens in virtual machines and on real metal. The same problem exists with sudo shutdown now, too.  I've lived with it in the past. The difference now is I don't want to just hit the power button because I might lose data. that looks an interesting book; they look good guys That looks an interesting book. ()   ‘Look’ can take an NP for its complement by OALD. Now what I’m wondering about is if a person/people could be a subject in the structure, for instance, “They look good guys.” If we can’t, does it have to be “They look like good guys”? How does the T-1000 see? In Terminator 2: Judgement Day, how does the T-1000 sense things? It's made of mimetic poly alloy; however, does it have a retinal scan camera like the T-X or T-800, or is just molecules that allow it to see what it does? In other words: let's say it is made out of nanobots. Would a group of these nanobots provide the capability of sight? Kind of like humans with cells that  make up the brain? I have always wondered whether James Cameron explained this on a special edition or if the topic was left up for insight from Terminator fans. From what I get from the whole mimetic poly alloy and some of the extra scenes, it seems to feel its way out to get information. Of course, it's very much not blind! Thoughts? Ubuntu 18.04 Dock mouse hover tooltips - how to disable? I need to know if it is possible and how to disable the Ubuntu 18.04 Dock tool-tips for mouse hover the pinned application. Vertical Text in Wpf TextBlock Is it possible to display the text in a TextBlock vertically so that all letters are stacked upon each other (not rotated with LayoutTransform)? "going to" vs "will" I know several questions were asked about the difference between "going to" and "will". Based on several answers (see, for instance, ,  and ), I understood that "will" is more spontaneous and "going to" is used with more planned actions.  So, it seems that everything is pretty fine. However, in , Kosmonaut has an answer in which he states: "Let's say that tomorrow you will walk your dog from 7 - 8 AM".  On the one hand, you probably planned to walk your dog long before and thus I should use "Let's say that tomorrow you're going to walk your dog from 7 - 8 am".   On the other hand, since I'm saying "let's say...", I'm deciding right now (thus, unplanned) that you will walk your dog. So, even though in this hypothetical situation you made a plan, I'm in a more spontaneous mood deciding right now that that's what you will do tomorrow, and, thus, I should use "will".  Which one (if any) of the above explanations is right? What happens to waves when they hit smaller apertures than their wavelenghts? I was wondering this for quite a long time now. Let's say you have a water wave (like ripples, not the ones you see during tsunamis) with wavelength 10 m. Imagine you put a boundary with an opening of 1 m. Will diffraction be observed? According to my research, no. But, then, what would be seen? My settlement was attacked by some Raiders and my defenses took care of them along with some of my NPC's. It seems like the Raiders didn't make it through to any of the buildings. What happens when they do?  Can you lose your items if you stored them in a house?  Can the NPC's die?  Do the enemies destroy your buildings? After upgrading to Ubuntu 17.10, Ubuntu Dock icons of some of my apps are not appearing. They appear properly in the dash. I verified the *.desktop files for the applications in /usr/share/applications and ~/.local/share/applications.  E.g. dock icon is not displaying for "Terminator". Check out the attached image.    This is how the icon for the same application appears in dash search:    Content of /usr/share/applications/terminator.desktop:  [Desktop Entry] Name=Terminator Comment=Multiple terminals in one window TryExec=terminator Exec=terminator Icon=terminator Type=Application Categories=GNOME;GTK;Utility;TerminalEmulator;System; StartupNotify=true X-Ubuntu-Gettext-Domain=terminator X-Ayatana-Desktop-Shortcuts=NewWindow; Keywords=terminal;shell;prompt;command;commandline; [NewWindow Shortcut Group] Name=Open a New Window Exec=terminator TargetEnvironment=Unity   Any pointers to debug this issue will be helpful. btrfs with multi devices fails automount in fstab I'm trying to automount btrfs  on 12.04-desktop-amd64. During bootsplash, fstab failed to mount btrfs and asked skip or manually prepare.  On the rootshell, btrfs device scan and mount -a could automount properly. It seems "btrfs device scan" was missing from bootstrap.  Where should I add the line to prepare bootstrap? How many $m\times n$ tables are there if they are subject to the following two conditions?    In each cell of the table we have $1$ or $-1$. The product of all the cells in any given row and the product of all cells in any given column equals $-1$.    I already know if Parity of $m$ and $n$ are different there is no such a table. what is the answer in general? Let $(a_n)_{n}\in\ell^2:=\ell^2(\mathbb{R})$ be a fixed sequence. Consider the subspace $$C=\{(x_n)_{n}\in\ell^2 : |x_n|\le a_n\text{ for all }n\in\mathbb{N}\}.$$  According to the book [Dunford and Schwartz, Linear operators part I, page 453] $C$ is compact in the $\ell^2$-norm, but there is no proof. How can I show that $C$ is indeed compact in $\ell^2$ ? Windows 10 update broke my MBR / GRUB I just spent the last 10 minutes trying to put a new set of grips on to my mountain bike handlebars..  The old ones came off fairly agreeably and I've got the new ones 80-90% of the way on, however, at this point both my hands are stinging from the aggressive twisting and pushing - I have tried gloves, but I feel it's best left till my hands don't hurt anymore.  Is there a tried and tested way of doing this? Or is there a knack to this that I am missing? Is this an instance of Priori Incantatem? During the climax of Fantastic Beasts and Where to Find Them, Tina briefly duels with Graves and their wands appear to lock with each other (see picture). I believe this also happens some other times in the later Harry Potter films other than Harry and Voldemort's duel in the graveyard during Goblet of Fire.    We know that this is definitely a case of Priori Incantatem for the graveyard duel in Goblet of Fire, as the wands are locked for a sufficient amount of time for past spells to begin showing themselves (significantly, showing the people Voldemort had murdered), but is this the case for the other instances of wands locking? If so, does that mean there are a lot more wands out there with shared cores, as it was my understanding this could only happen if that criteria was satisfied? Accessing Email logs using REST or SOQL URL to create a new Google document What is the current URL to create a new Google Document please? Since Google last updated their Docs the old URL does not work.