Let $f,g:[a,b]\to\Bbb{R}$ integrable functions. Let's dot each partition $P$ in two different ways, choosing in all the interval $[t_{i-1},t_i]$ a point $\xi_i$ and a dot $\eta_i$. Show that: $$\lim_{|P|\to0} \sum_{i=1}^n f(\xi_i)g(\eta_i)(t_{i}-t_{i-1})=\int_a^b f(x)g(x)dx$$  I need something as accurate as possible, because i really need to understand this question and i'm stuck at the very beginning.
I'm studying the Darboux definition of integrability, which I completely explained . There's an exercise that asks me to prove that the Darboux Integrability is equivalent to Riemann Integrability, but this Riemann integral is defined as the following:  It first defines a 'pointed' partition (I don't know how to say it in ensligh) by the following: a 'pointed' partition $[a,b]$ is a pair $P^*=(E,ξ)$, where $P=\{t_0, t_1, \cdots, t_n\}$ is a partition of $[a,b]$ and $ξ = (ξ_1, \cdots, ξ_n)$ is a list of $n$ chosen numbers such that $t_{i-1}\le ξ_i\le t_i$ for each $i=1,\cdots ,n$.  Now, the Riemann Integral is defined as:  $$\sum(f,P^*) = \sum_{i=1}^n f(ξ_i)(t_i-t_{i-1})$$  (I didn't understand the notation for the left hand side of the equation, by the way)  Finally, I'm asked to prove the following: given $f,g:[a,b]\to \mathbb{R}$ integrable functions, for the entire partition $P=\{t_0, \cdots, t_n\}$ of $[a,b]$ let $P^* = (P,ξ)$ and $P^{\#} = (P, η)$ be pointed partitions of $P$, then:  $$\lim_{|P|\to 0}\sum f(ξ_i)g(η_i)(t_i-t_{i-1}) = \int_a^b f(x)g(x) \ dx$$  I guess here I need to prove that the Riemann Integral of the product of two functions if the darboux integral of $f(x)g(x)$, but it seems too obvious, I just need to verify that $f,g$ are integrable, then their product is too, isn't it? I'm pretty sure this should be a hard question. Is there another interpretation that I'm missing?
Finding conditional distribution Let $X$ and $Y$ be independent $Exp(1)$-distributed random variables. Find the conditional distribution of $X$ given that $X + Y = c$ ($c$ is a positive constant).  this is my idea:  $$f_{X \mid X+Y}(x\mid c) = \frac{f_{X,X+Y}(x,c)}{f_{X+Y}(c) } $$  if this is the right definition then i might calculate the righthand side to obtain the distribution but Iam unsure Are there any guidelines for converting cooking recipes from Regular Ovens to Convection? I was looking to buy a Convection Oven and wanted to know if there are any general differences in baking methods between the two. Why is there not a simpler way to calculate the standard deviation? 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One example is this:  public void SetLabel(string text) {     Label.Text = text; }   In the example, Label is a control on a ASPX page. Is there a principle that can govern the decision (in this case) whether to make this a method or a property.  I'll accept the answer that is most general and comprehensive, but that also touches on the example that I have given. identity element of $(D_n ,\gcd)$ Let $x*y=\gcd(x,y)$ on $D_n=\{x\in \mathbb N:x |n\}$. I have to prove that $(D_n,*)$ is a commutative monoid and I have a problem when finding the identity element.  $x*e=x$  So $gcd(x,e)=x \to e=xk,k\in \mathbb N^*$ and $e \leq n$  A good $e$ is $0$ but it's not in $D_n$ 'find' utility does not output all files when using wildcards Running:  $find . -name *.exe   gives:  ./MakeItSo_1.2.2/MakeItSo_1.2.2/MakeItSo.exe ./MakeItSo.exe   but those are not the only exe files in the directory. For example, running  $find . -name ATLTester.exe   gives:  ./Debug/ATLDmoVexaTester.exe How to split strings into text and number? I am creating backup and after completion I got this..  Could not back up the following files. Please make sure you are able to open them.      /home/vinay/.cache/dconf     /home/vinay/.gvfs   I through terminal to open these file using  cd /home/vinay/.cache/dconf   It shows  bash: cd: /home/vinay/.cache/dconf: Permission denied   How can I get access to these files?  Is it fine to get permissions or leave as it is and not creating any backup for these files? What is "it" in the following sentence: It is clear that Bob likes doughnuts Continuous extension of a real function Related;    This is the theorem i need to prove;  "Let $E(\subset \mathbb{R})$ be closed subset and $f:E\rightarrow \mathbb{R}$ be a contiuous function. Then there exists a continuous function $g:\mathbb{R} \rightarrow \mathbb{R}$ such that $g(x)=f(x), \forall x\in E$."  I have tried hours to prove this, but couldn't. I found some solutions, but ridiculously all are wrong. Every solution states that "If $x\in E$ and $x$ is not an interior point of $E$, then $x$ is an endpoint of a segment of at most countable collection of disjoint segments.". However, this is indeed false! (Check Arthur's argument in the link above)  Wrong solution Q4.5;    Just like the argument in this solution, i can see that $g$ is continuous on $E^c$ and $Int(E)$. But how do i show that $g$ is continuous on $E$? What is wrong with my date regex? Waiting for user input with a timeout I have searched but apparently my google foo is weak. What I need is a way to prompt for user input in the console and have the request time out after a period of time and continue executing the script if no input comes in. As near as I can tell, Read-Host does not provide this functionality. Neither does $host.UI.PromptForChoice() nor does $host.UI.RawUI.ReadKey(). Thanks in advance for any pointers.  EDIT: Much thanks to Lars Truijens for finding the answer. I have taken the code that he pointed out and encapsulated it into a function. Note that the way that I have implemented it means there could be up to one second of delay between when the user hits a key and when script execution continues.  function Pause-Host {     param(             $Delay = 1          )     $counter = 0;     While(!$host.UI.RawUI.KeyAvailable -and ($counter++ -lt $Delay))     {         [Threading.Thread]::Sleep(1000)     } } How to select (highlight) text in Skype chat without using a mouse? Recently I started learning to use keyboard shortcuts almost for everything I'm doing on the laptop, but soon enough after I started, I bumped into the problem of being unable to highlight (select) simple text in Skype chats.  I did some research and I couldn't find any solution to my problem. Have I missed an already answered question that is the same like mine or if not what solution can you suggest in order to achieve the results I want without touching mouse or touchpad or any cursor moving device? How can I set desktop defaults for new users? How can I set up desktop defaults, much like most distributions do, for every new user? (ex. default wallpaper, apps, etc) Prove or disprove that $f(x) = \sin x/x$ is uniformly continuous over the interval $(0,1)$? I am thinking of $\lim_{x \to 0} \ {f(x)} = 1$ but I am confused as $x=0$ is not in the domain and also I want to write an $\epsilon$-$\delta$ proof. So any help is much appreciated!  Thanks! Setting PATH variable in /etc/environment vs .profile Where is the preferred place to set the PATH envvar?  ~/.profile or /etc/environment?  What is the case when PATH is set in both places? Is the end result a concatenation of both values set in those two places? Difference between Session and HttpContext.Current.Session What does "all the more" mean and how is it used? Here is a sample context:     As you might expect, the framework also provides some convenience   classes to make the task of writing custom validators all the more agreeable.   (Emphasis added.) newcommand vs. DeclareMathOperator As far as I understand, whatever can be done using DeclareMathOperator can also be achieved using newcommand, but not the other way around. If this is the case, what are the circumstances where DeclareMathOperator is the better choice? When should I take solely newcommand?  I found a question regarding , and I wonder what other aspects should be taken into consideration when it comes to choosing which method to use when defining math operators. Is the Ubuntu Phone available anywhere yet? I want start developing apps.  I am very impatient to start to develop on real ubuntu phone.