Let $f$ be a real-valued function defined on $\mathbb{R}$. Show that the set of points at which $f$ is continuous is a countable intersection of open sets.   Not sure where to start on this one... what would be the open sets?
I am trying to solve this exercise from Royden's 3rd edition.     The question is as follows: Let $f$ be a real-valued function defined for all real numbers. Show that the set of points at which $f$ is continuous is a $G_{\delta}$.    Let $$A_n = \{y : \text{there is a }~\delta_y \gt 0 : |f(s)-f(t)|\lt 1/n ~ \text{whenever}~ s,t \in (y-\delta, y+\delta)\}\;.$$     Then by the definition of open sets, $A_n$ is open.     To complete the proof, I need help in showing that $f$ is continuous at say $x$ if and only if $x\in \cap A_n$.     If $f$ is continuous at $x$, the there is a $\delta \gt 0$ such that $|f(x) - f(a)| \lt 1/n$ whenever, $x\in (a-\delta, a+\delta)$. so $x \in A_n$ son it must be in $\cap A_n$.  Thanks.
Why do PBX systems use -48 V? It seems common that PBX and other telephone hardware use a positive-ground power supply, where the "hot" line is at -48v.  What's the reason for that? RSA encryption with private key and decryption with a public key When using the RSA cryptosystem, does it still work if you instead encrypt with the private key and decrypt with the public key? What about in the case of using RSA for sender authentication? How often do ewoks eat sentients? In Return of the Jedi, the ewoks plan on eating Luke, Han and Chewie.     THREEPIO: I'm rather embarrassed, General Solo, but it appears you are to be the main course at a banquet in my honor.   At this point, the tribe had already met and befriended Leia, and had encountered the Imperials in the past, making it fairly likely that they knew humans were .  This raises the question of how often they ate sentients. Was it a one-off thing, to honor an extremely rare visit from a god, or was this fairly common? Set a default parameter value for a JavaScript function Solder does not stick to tip of soldering iron? Basic question here, but it's driving me insane!  The solder does not seem to stick to the tip of my soldering iron. After the iron heats up, the tip appears black (sometimes on all sides and sometimes on one side), then either it doesn't melt the solder at all, or it I fiddle around with it for a while; it melts the solder, but the solder rolls into itself into a ball and doesn't stick to the tip.  I've seen this many times while soldering before and usually it's solved by cleaning the tip on a soldering sponge or cooling it down and scraping the tip off with a blade, but I have scraped this particular one many times and still it's the same problem: it heats up and gets black, then the solder doesn't stick to it.  It doesn't help that this soldering iron has a very fine tip and I have to use it to solder very tiny chips and wires under a microscope (I'm just starting to learn soldering under microscopes).  Any ideas on what I can do?    PS: Is scraping the tip a bad idea? I have been doing it with my other bigger soldering irons and it seemed to work (at least in most cases) but I am not sure whether it is a good solution or not. How to understand Cauchy's proof of AM-GM inequality(the last step) The AM-GM inequality: $$a_1a_2\cdots a_n\leq\left(\frac{a_1+\cdots + a_n}{n}\right)^n$$  the trivial case: $a_1a_2 \leq \left(\frac{a_1+a_2}{2}\right)^2 $ is self-evident.  then cauchy use this fact repeatedly. He get: $$a_1a_2\cdots a_{2^m} \leq \left(\frac{a_1+a_2+\cdots +a_{2^m}}{2^m}\right)^{2^m}.\tag{$\ast$}$$  This is easy to understand, and natural, but next step:  he let  $$b_1 = a_1,\ b_2=a_2,\ \ldots,\  b_n=a_n,\  b_{n+1}=\cdots=b_{2^m}=\frac{a_1+a_2+\cdots a_n}{n}$$  replace $a_1, a_2,\ldots ,a_{2^m}$ with $b_1, b_2, \ldots b_{2^m}$ in $(\ast)$ and simplified both left and right. He get the answer. I can repeat his proof, but I don't know why Cauchy can got this idea. How does he got it? biblatex: \parencite in note field prints spurious ; with authoryear-icomp style Content Type field on NewForm.aspx I have a custom list and I am using content types to add items to the list. I am using OOTB forms for new and edit items. One field that I see on the EditForm.aspx is the one that I hope I get on the NewForm.aspx also and it is the Content Type field. It is shown in the following screenshot. Do you know how to add that field on the NewForm.aspx page also?    Thanks. Scenario:   Create SymLink: C:\Data\workspaces --&gt; C:\Users\****\Documents\workspaces Navigate to C:\Data\Workspaces in explorer Click and drag the the path from the address bar onto "Quick Access" in the navigation pane.   Expected Result:   Clicking on the shortcut takes you to C:\Data\workspaces   Actual Result:   Clicking on the shortcut takes you to C:\Users\****\Documents\workspaces   How do you make it so that the shortcut points to the correct path. (the SymLink)  Command Used to create the SymLink:  mklink /D C:\Data\workspaces C:\Users\****\Documents\workspaces   Updated with pictures:  Please note the workspaces symlink inside C:\Data:    When I double-click on the symlink it works correctly. (note the address bar.    When i pin it to the Quick access bar, however, it uses the address that the symlink points to and NOT the symlink address. White stains and stainless steel cookware I know this question has been asked before in some form or another, but I can't seem to find a definitive answer. I've looked on SA, chowhound, martha stewart, about.com, etc., and answers seem to vary. Answers that I've seen vary from using vinegar to washing using special stainless steel soaps (which I'm not very keen on using) to only using soft sponges and hot water.   The one thing I always do is wash my stainless steel with warm soapy water and a soft sponge. If there's burned food stuck on the cookware, I scrape it off with a wood pot scraper (lightly, of course). And I always dry with a soft cloth immediately after washing. Yet I still get stains!  I'd like to get a general consensus on how to prevent those white and rainbow colored stains as well as minimizing corrosion. Why the switch statement cannot be applied on strings? Compiling the following code and got the error of type illegal.  int main() {     // Compilation error - switch expression of type illegal     switch(std::string("raj"))     {     case"sda":     } }   You cannot use string in either switch or case. Why? Is there any solution that works nicely to support logic similar to switch on strings? How do you get a Media Type (MIME type) from a file using Java?  So far I've tried JMimeMagic &amp; Mime-Util. The first gave me memory exceptions, the second doesn't close its streams properly. How would you probe the file to determine its actual type (not merely based on the extension)? In the midst of attempting to prove that for a commutative ring $A$ with identity, and an ideal $I$ of $A$, $I = \operatorname{rad}(I)$, where $\operatorname{rad}(I) = \{x: x^m \in I, m &gt;0\}$, implies that $I$ is an intersection of prime ideals, I've run into a little bit of confusion.   An answer to another question on this site, , indicates a strategy that I am attempting to use. I know that if $x$ is an element of $A\setminus I$, then so too is $x^m$ for any positive integer $m$. I use this fact to consider a subset $X = \{x^m : m &gt; 0\}$, which is closed under multiplication, and use Zorn's lemma to find an ideal of $A$ containing $I$ maximal with respect to the constraint that said ideal, $J$, is disjoint from X.  I want to prove $J$ is prime by contradiction. I suppose that $J$ were not prime; this also means that $J$ is not maximal in $A$. Then there would exist two elements, $f$ and $g$, of $A$ so that $fg$ is in $J$, but neither $f$ nor $g$ is in $J$. Following the prompting of the aforementioned answer, I know that $J + \langle a\rangle$ and $J + \langle b\rangle$ would be ideals strictly containing $J$, and since J is the maximal ideal disjoint from $X$, both $J +  \langle a\rangle$ and $J + \langle b\rangle$ contain at least some elements of $X$ (I am tempted to say all of them, but I'm not entirely sure that's true, in case for example $\langle a\rangle$ started containing powers of $x$ at $x^2$, in which case it isn't necessarily true that $\langle a\rangle$ contains $x$ itself), and so both $\langle a\rangle$ and $\langle b\rangle$ must themselves contain elements of $X$.  And this is where I am stuck. I feel like there is something I should know about multiplicative subsets of rings that would help me arrive at the desired contradiction, but I can't remember what it might be. We currently have around 20 apex classes and 5 triggers which are not used in our production org. We want to delete them in our production instance. We dont have ant scripts ready yet to be used for our deployments and so we cannot use destructivechanges file. Currently our production deployments are done by our support team and they need documentation from us to do these deletes. Changesets do not allow to delete apex classes and triggers. So what is the best way to delete these apex classes and triggers from production? Split a string by a delimiter in python Nice looking progress bar in java Can Blender render pngs with the background transparent? I would like to render an animation as a series of .png images with transparent backgrounds for use in a video. Is this possible? Broken URL for data.stackexchange.com help page When using data.stackexchange.com I fail to remember if as [post link] is the correct alias to get the url for the post. Luckily the site has an  link. Unfortunately it is now broken. Can this be fixed? Proving that $\left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right) \lvert f(z) \rvert^2 = 4 \lvert f'(z) \rvert^2$ Given $f$ entire show that $$ \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right) \lvert f(z) \rvert^2 = 4 \lvert f'(z) \rvert^2 $$ I've come close to getting the exact answer by writing $f(z)$ as $u(x,y)+iv(x,y)$ and realizing the Laplacian of f is equal to zero. This leads to a lot of cancellations but I'm still making mistakes in my computations and I can't figure out where. Any help is appreciated. With the announcement of the detection of gravitational waves, questions about the implications proliferate.  Some relate to the possible existence of gravitons.  The analogous relationship between gravitons/gravitational waves and photons/electromagnetic waves is frequently mentioned.  The detection of individual photons required experiments of very low intensity light, yet their existence was inferred (prior to their actual detection) by Planck and Einstein (among others) using the properties of experimental black body radiation and the photo-electric effect.  If the prospects for detection of gravitons requires similar study of very low intensity gravitational waves, then those prospects are very dim indeed. My question: are there similar indirect "experimental" methods for inferring the existence of gravitons?