How to see that this set is countable? Let $f : \mathbb{R}\to \mathbb{R}$ be a monotone function and consider the set $S$ of all points $a\in \mathbb{R}$ such that   $$\lim_{x\to a^-}f(x)\neq \lim_{x\to a^+}f(x).$$  I want to show that this set is countable, but I'm not finding any way to do it. I thought one way would be to show that this set is discrete, so that it is countable. For that to happen, I would need to show that there is no convergent sequence $(x_n)$ of points of $S$ all different than $a$ whose limit is $a$.  My idea then is to suppose there is such a sequence $(x_n)$. Then $x_n\in S$ and $\lim x_n = a$ with $x_n\neq a$. In that case we have $\lim_{x\to x_n^-}f(x)\neq \lim_{x\to x_n^+}f(x)$.  Intuition says this should contradict $f$ being monotone, but I'm not finding a way to prove it.  So how can I finish this proof? Is my strategy correct? If so how do I finish it? If not, how should I prove this result?
How to show that a set of discontinuous points of an increasing function is at most countable I would like to prove the following:       Let $g$ be a monotone increasing function on $[0,1]$. Then the set of points where $g$ is not continuous is at most countable.     My attempt:    Let $g(x^-)~,g(x^+)$ denote the left and right hand limits of $g$ respectively. Let $A$ be the set of points where $g$ is not continuous. Then for any $x\in A$, there is a rational, say, $f(x)$ such that $g(x^-)\lt f(x)\lt g(x^+)$. For $x_1\lt x_2$, we have that $g(x_1^+)\leq g(x_2^-)$. Thus $f(x_1)\neq f(x_2)$ if $x_1\neq x_2$. This shows an injection between $A$ and a subset of the rationals. Since the rationals are countable, $A$ is countable, being a subset of a countable set.    Is my work okay? Are there better/cleaner ways of approaching it?
What's the deal with the phrase "could care less"?  Whilst growing up, I've always known people (parents etc) to use the phrase "couldn't care less", but I've also come across people who use the phrase "could care less" to mean the same thing (that is, "I do not care at all about that").  To me "couldn't care less" seems a lot more logical. If I couldn't care less about something, that implies I do not care about it at all. In contrast to this, to my mind the phrase "could care less" seems to indicate "I do care about this to some degree at least", but that's definitely not the meaning intended.  What's going on here? Is my (logical) analysis correct or in error? What does negative R-squared mean? Let's say I have some data, and then I fit the data with a model (a non-linear regression). Then I calculate the R-squared ($R^2$).  When R-squared is negative, what does that mean? Does that mean my model is bad? I know the range of $R^2$ can be [-1,1]. When $R^2$ is 0, what does that mean as well? prevent "action" if top of page I have a manual list of stuff that are sometimes separated by an hrule(or my own custom ruler) for visual purposes.  Sometimes the hrule ends at the top or bottom of a page which makes it useless.  Is there a way to prevent it from being added if it is at the top or bottom of a page?    MWE of problem:  \documentclass{book} \usepackage{tikz} \begin{document} \newcommand\rulesep{%   \par % make sure we end a paragraph   \vskip5pt % space above the rule   \leaders\vrule width \textwidth\vskip0.4pt % rule thickness is 0.4pt   \nointerlineskip % disable interline glue here   \vskip5pt % space below the rule } \def\dotikzcircle#1{\begin{tikzpicture}\draw(0,0)circle(#1);\end{tikzpicture}} \rulesep \noindent This is a test\noindent \rulesep \noindent End of test \rulesep \dotikzcircle3 \rulesep \begin{center}\dotikzcircle3\end{center} \rulesep \begin{center}\dotikzcircle3\end{center} \rulesep \dotikzcircle1 \rulesep \dotikzcircle2 \rulesep \dotikzcircle3 \rulesep \dotikzcircle4 \rulesep \dotikzcircle5 \rulesep  \end{document} What are the benefits of nickel-plated chains? Bicycle chains with higher model numbers (and slightly higher prices) are often nickel-plated or half-nickel-plated. What are the benefits of nickel-plated chains over non-nickel-plated chains?  How durable is the plating--does it wear off, or do its benefits last for the life of the chain? Loading PyQt5 modules from QGIS3/PyQGIS I am trying to talk to the PyQGIS API for use in a 3rd party application using the offical .dmg image provided on the QGIS homepage.  After setting PYTHONPATH to /Applications/QGIS3.4.app/Contents/Resources/python and initializing everything as stated in , I am facing the following error when running from PyQt5.QtGui import or from qgis import *:    ImportError: dlopen(/Applications/QGIS3.4.app/Contents/Resources/python/PyQt5/QtGui.so, 2): Library not loaded: @executable_path/../Frameworks/QtGui.framework/QtGui   Referenced from: /Applications/QGIS3.4.app/Contents/Resources/python/PyQt5/QtGui.so   Reason: image not found   Looks like I need to set some linkers or similar?   Side facts: - I have no problems when using the homebrew version of QGIS). - I am on macOS 10.14.6 (Mojave) This is a question that arose in the comments here:   I normally hide footnotes from the TOC by doing this:  \section[title]{title\footnote{This is a footnote.}}.   This works in a lot of document classes (memoir, article, ...), but not in all of them. amsbook, for example, uses the optional argument only for the running heads so the footnote mark still ends up in the TOC. So what is the 'universal solution'? Simple undivided plane UV map not behaving as expected How to disable the keyring for SSH and GPG? Duplicate application icons in Ubuntu dock upon launch I'm using Ubuntu 17.10 with GNOME shell and Numix-Circle icon theme.   When I hover over the Numix icon for Nightly Web Browser, which is pinned to the dock I get the title "Nightly Web Browser".      But when I launch the program, another icon pops up at the bottom, with the name "Nightly". I need to build a list of files present in a directory but also pass in a file containing a list of directories to exclude.  I've seen the below suggestion but I've had no luck with it when it comes to the type flag:  find $dir_name $(printf "! -name %s " $(cat $exclude_file))   Is there a way I can do it or will I need to create the entire list and then work through it removing ones that partially match ones in the exclude directory. Correct usage of ‘on’, ‘at’ and ‘in’ VB6 compiling via console (vb.exe) compatibility issue on 64bit (works on 32bit?) Would this hypothetical rule change ever matter? One of  is     104.3f If a player would both win and lose the game simultaneously, he or she loses the game.   . However, it seems to me like it "ought to be" the case that if there are only 2 players and  [A loses] ​ [B loses] ​ [B wins]  would occur simultaneously, then B should win, since  win-and-lose-simultaneously should be better than just-lose.  For that reason, it seems like the     Proposed Alternative : ​ ​ ​ If, simultaneously, all players would lose the game and      some players would win the game, then the players who would not win the game      lose the game, and the game is a draw between the remaining players. ​ Otherwise,      if a player would both win and lose the game simultaneously, he or she loses the game.   would make more sense. ​ However, I can't think of any situations in which that would matter.  Are there any possible situations in which my proposed alternative  would yield a different result from the actual rule? On what criteria should I evaluate domain registrars? Though I've been a web developer for a fair amount of time, I am going for the first time to buy a few domain names.  I have looked into the domains I'm going to buy and know that they're available, and I've been looking into which sellers to use.  After doing a lot of research, the main ones I'm considering are 1&amp;1, Namecheap, and Gandi.  The problem is, when continuing to research, I'm not really sure what makes one domain seller distinct from another.  I don't need much in the way of services--definitely not hosting, since I plan to use Heroku for that.  I mainly only need the domain itself and DNS management, as well as possibly SSL certificates and WHOIS protection.  Question:  What makes one domain seller different from another?  How can I go about evaluating which one is the best for me?  Note:  This question is not which domain seller is the best, but rather, what criteria can I use to evaluate them and rank one over another.  I'm trying to find out what makes one domain seller different from another, since they all seem to be pretty similar to me right now. Power series is locally normally convergent in its convergence radius $B(0,R)$ Real definition of "countable set" How do I reset all progress in the game "just shapes and beats"? The game  has a very unique intro that I would like to show my friend again for the first time. I'm not sure how to reset all my progress on this game to achieve this though. Any thoughts? Default poster size What is the typical size for a conference poster? Is it "ask the organizers", or can I safely assume that I'll get a default size, and that there will be enough space to hang it at the poster session? Trailing Sum Query How To Get Command Blocks Talking in Chat I need a way for a command block to say things in chat when it is powered, or more preferably whenever someone spawns into the world.