Let $f:\mathbb{R}\to\mathbb{R}$ be an integrable function. Show that if $\int\limits_a^bf(x)dx=0$ for all rational numbers $a&lt;b$, then $f(x)=0$ all most everywhere. Hint:  First prove $\int\limits_Af=0$ for $A$ an open set, then for $A$ measurable. My attempt: Let $A$ an open set in $\mathbb{R}$. Then  we can write $A=\bigcup\limits_{k}(a_k,b_k)$ where $\left\{(a_k,b_k)\right\}_{k=1}^{\infty}$ is a disjoint collection of open intervals with rational end points(Is this possible?) So $\int\limits_Afdx=\int\limits_{\bigcup\limits_{k}(a_k,b_k)}fdx=\sum\limits_{k=1}^{\infty}\int\limits_{a_k}^{b_k}fdx=0$ Then how should I use the result to for measurable $A$ and moreover, after doing so, does $\int\limits_{\mathbb{R}}f=0$ implies $f=0$  a.e? Appreciate your help
Suppose $f:\mathbb{R}\to\mathbb{R}$ is Borel measurable. I want to show that $\displaystyle \int_a^b f(x)\mathrm dx=0$ for all $-\infty&lt;a&lt;b&lt;\infty$ with $a$ and $b$ rational implies that $f=0$ a.e. I'm really not sure how to approach this problem; any help would be welcome.
I'm a 16-year-old male, and I have been interested in tabletop RPGs for a long time, especially D&amp;D. I sorta dove into D&amp;D really fast and hard by buying the Red box, Dungeon Master Kit, Monsters Vault, Players Handbook, Rules book, and even the Shadowfell Campaign not quite knowing what I'd get myself into. I've done tooonnsss of research and even watched live games on Youtube.   For the most part, I'm basically on my own, My friends are somewhat interested in it and want to play, but expect me to Teach/Dungeon Master the game for them, I'd really wish to do this, but need more insight and expertise. I've played the solo Campaign in the Red box and even thought about joining a local group from a popular gaming store in Phoenix, AZ called Imperial Outpost. But I'm not sure if they would take a newbie like myself. Any advice? How to simplify the summation kk! without using induction? Don't count non-printing characters, HTML comments or other content not displayed toward minimum character requirements Compact space and Hausdorff space A continuous map from a compact space to a Hausdorff space is closed. Why this is true?  Help me please I want to learn why this is correct. I am running Ubuntu 14.04 on think-pad T450.  When using an external monitor with higher resolution than the in-built laptop screen, the text of dialog boxes are getting messed up.  Looks like I need to replace the Graphic card driver. Anyone found a solution to this?  Updated to add requested information:  lspci -nnk | grep '\[03' -A2   gives the following:  00:02.0 VGA compatible controller [0300]: Intel Corporation Broadwell-U Integrated Graphics [8086:1616] (rev 09)     Subsystem: Lenovo Device [17aa:5034] Kernel driver in use: i915   Below is a screenshot, see how the letters are messed up on the screen. Trisect a quadrilateral into a $9$-grid; the middle has $1/9$ the area I wonder if it is possible to consider $\mathbb R$ as a Cartesian product $X\times X$ for some set $X$. From the point of view of the dimensionality, there are spaces with a Hausdorff dimension $1/2$ (sort of Cantor sets), but I guess there are other problems in this construction  unrelated to the dimension.  Edited: replying on the comment by Asaf. I want that for all $r\in\mathbb R$ there exists a unique representation $r=\langle x,y\rangle$ where $x\in X$ and $y\in X$. Also, what if we want to have $X$ to be a topological space and $h:X\times X\to\mathbb R$ to be a homeomorphism? $a_1x_1+a_2x_2+a_3x_3+...+a_nx_n=$ is called a linear equation because it represents the equation of a line in an n dimensional space. So "linear" comes from the word "line".Basically there should not be any higher power of x failing which the graph of the function will not be a straight line.  simillarly  $a(x)y+b(x)y'+c(x)y"+d(x)y'''+...+q(x)=0$ is also called linear differential equation because all the derivatives have power equal to 1 which is similar to the above definition of a linear equation.  A function f is called linear if: $f(x+y)=f(x)+f(y)$ and $f(cx)=cf(x)$. Here c is a constant. In this definition of linearity of function "$f$" what does the word linear means? How does it relate to a straight line?  Finally what does the term linear means in case of linear vector spaces? Where is the reference to a straight line?  So, whether linear is just a word used in different contexts? Does it have different meaning in different situation? Or linearity refers to some relation to a straight line? At Least please explain how the linearity of function f and linear vector space relate to the equation of a line. I've recently read the Yahoo manifesto . They recommend to put the JavaScript inclusion at the bottom of the HTML code when we can.  But where exactly and when?  Should we put it before closing &lt;/html&gt; or after ? And above all, when should we still put it in the &lt;head&gt; section? How do I find out what tool a shortcut is triggering? I used to have problem with connecting two objects, because when I smooth the final object, there was a darker line on the connected parts (for example when I was making head, I've had eyes darker this way). Then I found out that I can get rid of it by hitting Ctrl+N, but I don't realy know what it does. I can just see that the lines are gone. So, what does Ctrl+N do? How would I remove the border from an iframe embedded in my web app? An example of the iframe is:  &lt;iframe src="myURL" width="300" height="300"&gt;Browser not compatible.&lt;/iframe&gt;   I would like the transition from the content on my page to the contents of the iframe to be seamless, assuming the background colors are consistent. The target browser is IE6 only and unfortunately solutions for others will not help. Why won't Smart UV Project automatically unwrap a simple chair? I created a very simple chair mesh and I would like to unwrap it so that I can map a texture. I would like to have Smart UV Project unwrap it automatically, but whether I do it that way or try manually, I am unable to unwrap the model.   Is there something I'm missing?    I tried a few guides, but I'm not sure where I'm going wrong.  My File: Who is allowed to invite people to a Minecraft Realm? Why are some applications so hard to find in the Ubuntu Software Centre? Some applications don't pop up in the search even when you type in their name. Take for instance Sushi. When you type in "sushi" into the search box, it fails to throw up the application. Only when you type in "gnome-sushi" does it appear. This is despite the fact that the application has "sushi" by itself in the application description. There are other applications which don't spring to mind right now, but I've discovered before. What is the difference between building from source and using an install package? I was wondering: when installing something, there's an easy way of double clicking an install executable file, and on the other hand, there is a way of building it from source.   The latter one, downloading a source bundle, is really cumbersome.  But what is the fundamental difference between these two methods? What the #$@&%*! is that called? Is there a name for the use of symbols in place of curse words, for example #$@&amp;%*!? Nautilus bookmarks are missing since upgrade to 13.04 My bookmarks are not shown anymore after upgrading to 13.04 Raring ringtails and I can't add any new bookmarks. The function is not shown and the shortcut Ctrl+D isn't working either. Is this functionality just gone for some reason (as the split screen with F3) or is there something wrong with my upgrade?  Thanks for any hints. Showing that $f$ is linear function if $\forall z \in \mathbb{C}$, $|f(z)| \leq 1 + |z|$. Let $f$ be an entire function that satisfies $|f(z)| \leq 1 + |z|$ for all $z\in \mathbb{C}$. Show that $f(z) = az +b$ for fixed complex numbers $a,b$.    The hint tells us to try and use Cauchy's Integral Formula on an arbitrary circle.    This is my attempt: Consider an arbitrary large circle with centre $0$ and let $\Upsilon$ be the contour around this circle.    Then $$\int_{\Upsilon} \frac{f(z)}{z^{n+1}} \mathrm{d}z = \frac{2\pi i}{n!}f^{(n)}(0).$$    Note that $$\int_{\Upsilon}\frac{f(z)}{z^{n+1}} \leq \int_{\Upsilon}\left|\frac{f(z)}{z^{n+1}} \right|\leq \int_{\Upsilon} \frac{1}{|z^{n+1}|} + \frac{1}{|z|^n}$$ and the right hand side is $0$ by Cauchy's Integral Formula. I'm not sure if the first equality is valid, and not sure where to go from here. I'm going to university in October and thought I'd have a go at a few questions from one of their past papers. I have completed the majority of this question but I'm stuck on the very last part. In honesty I've been working on this paper a while now and I'm a bit tired so I'm probably giving up earlier than I usually would.  I won't write out the full question, only the last part:     Let $$S_r(n) = \sum_{m=0}^n\ (-1)^m m^r {n \choose m}$$ where r is a   non-negative integer . Show that $S_r(n)=0$ for $r&lt;n$. Evaluate   $S_n(n)$.   I have shown that $S_r(n)=0$ for $r&lt;n$ by taking $(1+z)^n= \sum_{m=0}^n\ z^m {n \choose m}$, letting $D_r(f(z))=z\frac d{dz}(z\frac d{dz}...(\frac d{dz}(f(z)))...)$ where $z\frac d{dz}$ is applied $r$ times and applied it to both sides.  The left hand side gives a polynomial, degree n which has factor $(1+z)$ for all $r&lt;n$ and the right hand side gives $\sum_{m=0}^n\ z^m (m)^r {n \choose m}$. Setting $z=-1$ yields the required result.   There is some build up to this, so I'm fairly certain that this was the intended method.  I'm stuck however on the very last part. I have tried finding a form for $D_r((1+z)^n)$, but I'm fairly sure that this isn't the correct approach, as the wording of the question implies that $S_n(n)$ needs to be considered separately.  I'm surprised that I didn't find that this question had already been asked, so apologies if it has been.  Thank you. Why is there an emission of gamma rays? When a spontaneous radioactive reaction happens, there is an emission of gamma rays (in most cases) What causes this emission?