Prove that the function : $f(x) = x$ when $x$ is rational and $f(x) = 1 - x$ when $x$ is irrational is continuous at $x = \frac{1}{2}$  To prove the continuity it is necessary to prove that for every $\epsilon$ there exists a $\delta$ such that $|f(x) - f(\frac{1}{2})| &lt; \epsilon$ whenever $|x -c| &lt; \delta$ $- (1)$ Now if we choose $a  \in (\frac{1}{2} - \delta, \frac{1}{2} + \delta)$ and also $a $ is rational and choose $\epsilon &lt; |a - \frac{1}{2}|$ then condition in $(1)$ is not satisfied. How to move from here?
Prove that the function : $f(x) = x$ when $x$ is rational and $f(x) = 1 - x$ when $x$ is irrational is continuous at $x = \frac{1}{2}$  To prove the continuity it is necessary to prove that for every $\epsilon$ there exists a $\delta$ such that $|f(x) - f(\frac{1}{2})| &lt; \epsilon$ whenever $|x -c| &lt; \delta$ $- (1)$ Now if we choose $a  € (\frac{1}{2} - \delta, \frac{1}{2} + \delta)$ and also $a $ is rational and choose $\epsilon &lt; |a - \frac{1}{2}|$ then condition in $(1)$ is not satisfied. How to move from here?
How do I find out which program and process ID accesses a given IP address in Windows? Is it possible to catch, which program is accessing a specific IP address?   I found my computer is flooding and wish to check addresses one by one.  May be it is possible to set up some audit for this? I have been trying to solve this problem for quite a while. I am still unsure of whether any of the avenues I have pursued have been of any use. Any advice will be much appreciated. Question: Let $V$ be a finite-dimensional inner product space, and let $E$ be an idempotent linear operator on $V$. Prove that if $EE^* = E^*E$, then $E$ is self-adjoint. (This is essentially exercise 5(a) in sec. 80 on p.162 of Paul R. Halmos, Finite-Dimensional Vector Spaces, but Halmos didn't assume that the dimension of $V$ is finite.) I don't know so much about coordinate systems...  In my office we use to deal with spatial data coming from archaeological sites. Each site has its own x-y-z coordinate system (GCS). Three simple ortogonal cartesian axis. In the last years we have been managing this spatial data through GIS software (ArcGIS), without using specific coordinate system (just leave it as "undefined")  I'd like to know if there exisits any GCS designed to deal with such datasets using simple cartesian orthogonal axis, without grid distortions of the typical GCS. In addition, I'd like to know if this system is suitable for using it in an online mapping application.  By the way, we manage 2D (ArcMap) and 3D (ArcScene) environments and work with "mm" as length base unit.   If such a thing doesn't exists, maybe someone knows how to create it. What is the ancestral form of language used in the fable that David was studying during the flight? David studies linguistics during the flight to LV-223. We get to listen and read a fable. Is it written in a real language? What is the name of the language? Can the fable be translated?  Approximate quotation:     hjewis jasma hwaelna nahast akwunsez dad'kta, tam ghermha vagam ugenthe I dug a large room to specifications, 3 blocks tall and 16 blocks wide as well as long. I dug it at 10 levels above bedrock, in a location {(-240, 0) -> (-224, 16)} specified by  slime finder app. I am definitely more than 24 blocks away and less than 32 away from the spawn room, as specified by the wiki .  Yet, no slimes. I have been sitting in my location patiently listening for the moist slapping sounds of slime. I have gone in to the room every time I heard a phantom slap but still nothing.  Am I just being dumb somehow or do the minecraft lords just hate me?  If it helps/anyone cares here's my seed: 6405460532872068998 ModernCV Remove horizontal line in sections I want to write my cv with moderncv-Classic style. Can you tell me, how I can remove the horizontal line in the left of each section? Sentient Bomb in a city on the moon? I would like to know the title of the short story that I believe takes place on the moon.  The setting is a mall or some other very public location and the main character is a police officer.  She needs to talk a sentient bomb out of exploding.  I know there was the John Carpenter film "Dark Star" with a similar plot line but I'm trying to identify the short story and the author. How much electronics and other valuables can I bring duty-free when going to India? I have a quick question about updating my references from Bibtex.  I am using TexStudio as my Latex editor.  I have to make changes to some of the references.  I put the changes in the "references.bib" file.  I then compile the Bibtex and the main document I'm working on.  Sometimes the changes show up, and sometimes they don't, and I haven't figured out why. I have even gone as far as trying to delete the references by changing the filename, but the original references still appear.  Here's what I put in my main document:   \newpage %Puts in References  \bibliographystyle{unsrt}  \bibliography{references}   My references.bib file shows all the changes I have made.  Any suggestions are appreciated! Why is a 1" sensor actually 13.2 × 8.8mm? I'm wondering how sensor sizes are calculated. The Sony RX100 reportedly has a sensor size of 1" which is 13.2 x 8.8 mm according to the press release and various sites.  How does that work? That's a pretty small inch.  The Wikipedia page on  has this chart:   which doesn't really help. (The 1" format is labelled Nikon CX. Chart by Moxfyre under the CC-BY-SA license.)   If full frame/35 mm refers to the horizontal size, what does 1" refer to? I'm trying to set an RSA key as an environment variable which, as a text file, contains newline characters.  Whenever I attempt to read from the file and pass it into an environment variable, it will just stop at the newline character on the first line. How can I prevent this? Why is my Canon 300D not switching on despite changing both batteries and the CF card? Is there a way to determine how many digits a power of 2 will contain? Is there a direct way to determine how many digits a power of 2 will contain without actually performing the multiplication?  An estimation would help as well if there is no absolute solution.  EDIT: In both decimal and binary bases. Unable to Telnet into port 25 on CentOS PostFix Server How to programmatically send SMS on the iPhone? Does anybody know if it's possible, and how, to programmatically send a SMS from the iPhone, with the official SDK / Cocoa Touch? Weird line in my mirrored mesh was following a tutorial where I stumbled into this weird line in my mirrored mesh (sorry for the feet lol)      You can also download the blender file Epson LX-310 not working after Cumulative Update 2017-11 Windows 10 I have Windows 10 Pro x32 and an Epson LX-310 printer that was working. Yesterday, Windows 10 automatically updated with 2017-11 Cumulative Update for Windows 10.   Today, I cannot print. The printer status is on. When I click "Print a test page", it quickly says the test page has been sent but the printer does not print it.   My printer is connected to a USB port. What wrong with my Windows 10? Recursive search in nautilus 3.6 completely breaks my keyboard navigation. I often navigate network volumes, and typing to get directly to a folder and clicking enter was very quick. Now typing removes everything in the view and imposes a 30 second delay which is completely useless.  If it showed whats in the current directory immediately, and then searched it would be fine, but as it is now I just want to turn it off. I couldn't find anything in dconf-settings. Anyone know how to disable it? Does $IJ=IK\implies J=K$ always hold for integral domain and finitely generated nonzero ideal $I$? Let $R$ be a commutative integral domain, $I,J,K$ three ideals of $R$ with $I\neq (0)$ being finitely generated. Then does $IJ=IK$ imply $J=K$?   With Nakayama lemma, I can prove it if one of $J$ and $K$ equals to $R$. And I also know it holds when $R$ is a Prüfer domain or $I$ is singly generated. If you hash a string using SHA-256 on your computer, and I hash the same string using SHA-256 on my computer, will we generate the same value? Does the algorithm depend on a seed (so we'd both need the same seed) or some other such parameter?  edit:  To clarify, by 'string' I meant 'the same byte input', which as the comments and @ispiro's answer point out, may be different for the same character string depending on the encoding.