I understand the proof that shows that if $f$ is monotone on an interval, then it has at most countably many jump discontinuities. I feel like a similar idea also allows us to show that any function can have at most countably many jump discontinuities. Here is my attempt at a proof.  If a jump continuity of $f$ exists at $x$, this means that $f(x^-)$ and $f(x^+)$ exist, but $f(x^-)\neq f(x^+)$. The existence of left and right hand limits means that for any $\epsilon &gt; 0$, we can find a $\delta &gt; 0$ such that if $y\in (x,x+\delta)$, then $|f(x^+) - f(y)| &lt; \epsilon $ and if $y\in (x-\delta, x)$, then $|f(x^-)-f(y)| &lt; \epsilon$. Thus, the only discontinuity (jump or otherwise) in the interval $(x-\delta, x+\delta)$ is at $x$.   I believe that this means that the jump discontinuities only exist within disjoint open intervals, of which there are at most countably many. If there does appear to be an issue at the boundary of the interval, for example, in a case where $x&lt;y$ are both jump discontinuities and $x+\delta_x = y-\delta_y$, then we can make $\delta_x$ or $\delta_y$ a bit smaller to prevent this overlap.   Is this proof correct, or am I missing something? This statement seems a bit strong, especially since I've only heard a similar statement for monotone functions on an interval. If it's not true, can someone give a counterexample?
I know that Dirichlet function has uncountable many discontinuities. I think they are removable, because the discontinuities can be removed by redefining the function values of the rational numbers as 0.  So Dirichlet function is a function that has uncountable many removable discontinuities, then my question is can we construct a function with uncountable  many jump discontinuities? If not, how do we prove it is impossible? Thank you.  An odd but similar question is can we have a function that has uncountable many infinite discontinuities?
What happens to Jen when she jumps off the cliff? What happens to Zhang ziyi (Jen) in  when she jumps off the cliff .. she dies? becomes immortal? Just before jumping what does she want Lo to become - a master?  Why does she jump off, guilt? A quick search for ESTA application returned many results. According to , the official site to apply for an ESTA is .  Are the others scams, legitimate third party resellers or "added value" services?    (defunct - domain is for sale circa 2020)     Can anyone confirm either way?  Updated 2020: The above have better "not affiliated with the U.S. Government" warnings now, but beware there are newer more sneaky sites starting all the time:        And fortunately more official guidance about this issue: Redirecting from http to https in Apache This is a  about redirecting from http to https in Apache Related:     I have an Appache web server which serves both http://example.com/ and https://example.com/. I want to redirect all http requests to https. Currently I'm using this .htaccess rule to redirect http requests to https. RewriteEngine On RewriteBase / RewriteCond %{HTTPS} off RewriteRule (.*) https://%{HTTP_HOST}%{REQUEST_URI}   It's working as expected for example.com but the same rule is not working for sub links, when I access existing links, like example.com/about it will still load in http, no redirection happens for existing links. How can I make Apache redirect all http requests to https? Many times I see that URL extensions are hidden (e.g. .html, .php), but not all websites do this.  Why do webmasters hide the extension? Is it for security, for the URL to look cleaner, or for some other purpose? Posting images in chat using drag and drop I think it will be a good idea if there is a drag and drop image posting in chat. User needs to follow many steps to upload an image using upload button. Print total free memory from `free -m` free -m currently puts out something like this.     I would like to know using grep/awk how we can get the total free amount. i.e. 9083 Last night I was updating my Ubuntu 10.04 to either 11.04 or 12.04 (I'm not entirely sure which one it was, but I think it was 11.04) and I left it overnight to finish. When I came down in the morning, I noticed that the computer hadn't barely begun installing the updates! The update thing said that my computer was on the installation step, but it was still preparing the libc bin (or whatever its called) . It's like the first part of the step installing the upgrades. So, I stupidly restarted the computer, and when it rebooted, I got five options on a purple screen:  Ubuntu, with Linux 3.0.0-28-generic Ubuntu, with Linux 3.0.0-28-generic (recovery mode) Previous Linux versions Memory test (memtest86+) Memory test (memtest86+, serial console 115200)   Long story short, not matter which way I enter Ubuntu to try and reinstall the updates using the root thing (it's like terminal) after I enter sudo apt-get update it basically said it failed to fetch every url link, and it then says:  W: Some index files have failed to download. They have been ignored, or old ones used instead. W: Not using locking for read only lock file /var/lib/dpkg/lock E: dpkg was interrupted, you must manually run 'sudo dpkg --configure -a' to correct the problem.   So, I entered sudo dpkg --configure -a, and it says:  dpkg: error: unable to access dpkg status area: Read-only file system"   I'm left like wtf, and I really just some guidance here. I just got this laptop, and I can not afford a new one if I broke this one. How to prevent nginx redirecting from HTTPS to HTTP on AWS? I have a website served by nginx behind an AWS ELB Load Balancer. Only HTTPS is enabled on the load balancer.  Requesting individual files, or directories with a trailing slash, work fine. However, requesting directories without a trailing slash don't work.  The reason is, when I request the directory without the trailing slash, nginx does a redirect to the path with the trailing slash (that's OK), but it also changes from HTTPS to HTTP. The load balancer is configured to only allow HTTPS, so that doesn't work (timeout).  On the nginx logfile, I can see that the request reaches nginx, and that it's nginx that responds with the 301 Permanent Redirect (so it's not e.g. a problem with the load balancer setup).  10.100.10.15 - - [24/Nov/2017:15:41:08 +0000] "GET /admin HTTP/1.1" 301 178 "-" "Wget/1.18 (darwin16.0.0)"   When I request the URL via curl I see the redirect:  $ curl -v https://example.com/admin *   Trying 1.2.3.4... * TCP_NODELAY set * Connected to example.com (1.2.3.4) port 443 (#0) * TLS 1.2 connection using TLS_ECDHE_RSA_WITH_AES_128_GCM_SHA256 * Server certificate: example.com * Server certificate: Amazon * Server certificate: Amazon Root CA 1 * Server certificate: Starfield Services Root Certificate Authority - G2 &gt; GET /admin HTTP/1.1 &gt; Host: example.com &gt; User-Agent: curl/7.54.0 &gt; Accept: */* &gt;  &lt; HTTP/1.1 301 Moved Permanently &lt; Date: Mon, 27 Nov 2017 09:19:05 GMT &lt; Content-Type: text/html &lt; Content-Length: 178 &lt; Connection: keep-alive &lt; Server: nginx &lt; Location: http://example.com/admin/ &lt; X-UA-Compatible: IE=Edge &lt;  &lt;html&gt; &lt;head&gt;&lt;title&gt;301 Moved Permanently&lt;/title&gt;&lt;/head&gt; &lt;body bgcolor="white"&gt; &lt;center&gt;&lt;h1&gt;301 Moved Permanently&lt;/h1&gt;&lt;/center&gt; &lt;hr&gt;&lt;center&gt;nginx&lt;/center&gt; &lt;/body&gt; &lt;/html&gt; * Connection #0 to host example.com left intact   My nginx config file is just   server {     root /var/www; }   The /etc/nginx/nginx.conf is .  I have tried server_name_in_redirect off but it did not make any difference.  I would like to avoid putting the hostname in the config file, because this is packed into a Docker image which is then deployed on different hosts (QA, Prod, etc.).  I would like nginx to do this redirect, but stay on HTTPS. What can I do? When an air-filled balloon is released without its opening tied up, it moves in a circular path rather than a straight line. Why is that? Management of fruit bowls on the Enterprise D I have noticed that guest quarters on the Enterprise D have fruit bowls on their tables.  A prominent instance is the fruit bowl in Berlinghoff Rasmussen's quarters in "".  The first thing that Rasmussen does when Data takes him to his guest quarters is take a piece of fruit and shove it in his pocket.  Who replicates fruit for these fruit bowls before guests arrive?  More generally, who removes unused fruit and tidies quarters after guests leave?  Is there a ship hospitality staff, perhaps including Guinan's wait staff in Ten Forward? NMI watchdog: BUG: soft lockup - CPU#2 stuck for 23s! [plymouthd:305] I get this error message   NMI watchdog: BUG: soft lockup - CPU#2 stuck for 23s! [plymouthd:305]    on the startup of my system (even with live-cd).  My PC is a dell inspiron 7559 with intel graphic and nvidia.  UPDATE - this issue also happens when attempting to install ubuntu 18.10 using LiveUSB Cannot vote to close immediately after edit Android app - can I see interesting questions? On the website the default list of questions displays interesting tab (which I believe has questions based on my highest scoring tabs). That's very convenient and allows me to check new questions that are relevant for me easily.   Can the same view be added to the mobile app? Solving "mv: Argument list too long"? I have a folder with more than a million files that needs sorting, but I can't really do anything because mv outputs this message all the time  -bash: /bin/mv: Argument list too long   I'm using this command to move extension-less files:  mv -- !(*.jpg|*.png|*.bmp) targetdir/ What's the best way to take a segment out of a text file? Aligning multiple points to line in QGIS? I would like to align multiple points to a line, or lines within a layer, using a specified tolerance or buffer around the line objects. Please refer to the sample sketch attached.  For the sake of this example, the points closest to the line in the BEFORE picture are within 5 map units of the line, while the outermost points are over 10 map units away. I would like to snap the closest points onto the nearest line, using a tolerance of 5 map units to achieve the result in the AFTER picture. Why aren't search engines indexing my content? This is a general, community wiki  question and answer pair intended to address any questions   concerning the reasons a site or specific site contents do not appear   in search engine results.      If your question was closed as a duplicate of this question and you   feel that the information provided here does not provide a sufficient   answer, please open a discussion on .   My site (or specific pages on my site) is not appearing in search engine results.  Why isn't my content indexed and what can I do about it? Is  && statement() the same as if() statement()? How to insert an item into an array at a specific index (JavaScript)? I am looking for a JavaScript array insert method, in the style of:  arr.insert(index, item)   Preferably in jQuery, but any JavaScript implementation will do at this point. I downloaded Ubuntu 14.04 (64 bit) lts. I made a bootable usb stick. But I keep receiving this error message upon booting:  SYSLINUX 4.04 EDD 20110518 Copyright (C) 1994-2011 H. Peter Anvin et al Unknown keyword in configuration file: gfxboot.c32: not a COM32R image