Let $A\subseteq \mathbb{R}$, function $f\colon A\to \mathbb{R}$ continuous at $a\in A$, and $g\colon A \to \mathbb{R}$ is discontinuous at $a\in A$.   Prove or disprove that $f+g$ is discontinuous at $a\in A$.   A proof by definition is what I need. But if ones try with the other way that's okay. Help me.  Thank you.
Let $x_0 \in \mathbb{R}$ and $f,g: \mathbb{R} \to \mathbb{R}$  such that $f$ is discontinuous at $x_0$ but $g$ is continuous at $x_0$, then $f+g$ at $x_0$ is...   My approach: After some graphical attempts to construct such a pathological case as given above I claim that the described scenario isn't possible. That is $$f \text{ disc. at } x_0 \text{ and } g \text{ cont. at } x_0 \implies f+g \text{ disc. at } x_0 $$  Proof: I was trying to come up with a contradiction, because if they happen to work out I find these kind of proofs to be most satisfying.   Assume $f+g$ is continuous at $x_0$ that is $$\forall \epsilon &gt; 0 ,\exists \delta_1 &gt; 0 : \forall x \in \mathbb{R}, |x-x_0| &lt; \delta_1 \implies |f(x)+g(x)-f(x_0)-g(x_0)| &lt; \epsilon  $$  Since this statement must be true for all $\epsilon &gt; 0$ I could choose $\epsilon:=1$. Next I know that also $g$ is continuous at said point $x_0$, thus  $$\forall \epsilon &gt; 0, \exists \delta_2 &gt; 0 : \forall x \in \mathbb{R}, |x-x_0| &lt; \delta_2 \implies |g(x)-g(x_0)| &lt; \epsilon $$  and I have that $f$ is NOT continuous at $x_0$ therefore: $$\exists \epsilon^* &gt; 0, \forall \delta&gt;0: \exists x \in  \mathbb{R}, |x-x_0| &lt; \delta \wedge |f(x)-f(x_0)| \geq \epsilon $$  My problem is to bring all these three statements nicely together. Mainly I get confused by the quantors. In order to match all three statements I assume I have to choose $\epsilon &gt; 0$ such that the last statement is met, so say $\epsilon:= \epsilon^*&gt;0$, so statement 1 and 2 are still true for said $\epsilon$  I also need to choose $\delta$. Lets say $\delta = \min{(\delta_1,\delta_2)}$ now I should be able to work with all three statements at the same time right? But then I stumble with my argumentation.  Let $|x-x_0|&lt; \delta$, then we obtain $$ |f(x)+g(x)-f(x_0)-g(x_0)| \leq |f(x)-f(x_0)| + |g(x)-g(x_0)| &lt; - \epsilon + \epsilon =0 $$ which means that $|f(x)+g(x)-f(x_0)-g(x_0)| &lt;0$ and I hope that this is a contradiction because $|.| \geq 0$. Please comment (or answer) if I am right or absolutely on the wrong path. I'd also appreciate it if you criticize my formulation.
If $\gcd(a,b)=1$, how can I find the values that $\gcd(a+b,a^2+b^2)$ can possibly take? I can't find a way to use any of the elemental divisibility and gcd theorems to find them. Showing $(a^{2}+2)(b^{2}+2)(c^{2}+2)\geq 9(ab+bc+ca)$ Help us test the alpha version of our Android app We are in the early stages of development of the official Stack Exchange Android app and we need your help with testing. To set a few expectations: there will be bugs, crashes, and missing features, all of which will be fixed in .    We are initially targeting Android devices running  (Android 4.1 or higher) starting with phones (up to ~5 inch) and eventually moving our way to tablets.    To register for alpha testing .  After you register we will send you an invite to a private Google community where you can find a link to download our app directly from the  store. The invitations will be sent out in waves, so if you don’t get an invite immediately, be patient.  Crash reports will be automatically reported back to us using . Bugs or feature requests should be reported here on meta and tagged . Please do not use the Google community to report bugs.  As always, we welcome your feedback and are committed to building the best mobile app possible for browsing the Stack Exchange sites. I want to put four figures in a single row. The code I am trying to use is taken from .  This puts three figures in a row. When I add a fourth one, I get three images in row and the last one appears in the second row. But I want all of them in one row. Here is my code.  \documentclass[12pt,twoside]{report}% Use this line for the print version of the thesis   \usepackage{graphicx} \usepackage{caption} \usepackage{subcaption} \begin{document}  \begin{figure}         \centering         \begin{subfigure}[b]{0.3\textwidth}                 \centering                 \includegraphics[width=\textwidth]{gull}                 \caption{A gull}                 \label{fig:gull}         \end{subfigure}%           \begin{subfigure}[b]{0.3\textwidth}                 \centering                 \includegraphics[width=\textwidth]{gull}                 \caption{A gull2}                 \label{fig:gull2}         \end{subfigure}%          ~ %add desired spacing between images, e. g. ~, \quad, \qquad etc.           %(or a blank line to force the subfigure onto a new line)         \begin{subfigure}[b]{0.3\textwidth}                 \centering                 \includegraphics[width=\textwidth]{tiger}                 \caption{A tiger}                 \label{fig:tiger}         \end{subfigure}         ~ %add desired spacing between images, e. g. ~, \quad, \qquad etc.           %(or a blank line to force the subfigure onto a new line)         \begin{subfigure}[b]{0.3\textwidth}                 \centering                 \includegraphics[width=\textwidth]{mouse}                 \caption{A mouse}                 \label{fig:mouse}         \end{subfigure}         \caption{Pictures of animals}\label{fig:animals} \end{figure} \end{document}   As I have mentioned above, it works for three figures.  But I can not use it for four figures. What should I change?   What does  width=\textwidth mean?   I changed    \begin{subfigure}[b]{0.3\textwidth}   to  \begin{subfigure}[b]{0.2\textwidth}   but it didn't help.  Then, I changed     \includegraphics[width=\textwidth]{mouse}   to  \includegraphics[width=0.5\textwidth]{mouse}   Still no solution. Catenatives followed by infinitives and gerunds I recently updated my profile bio (from Arqade) and clicked 'Save and copy changes to all public communities'. This appears to have broken/reset my profile picture on all but Arqade. Note that I did not update my profile picture as part of my changes. Arqade:  Meta SE for comparison:  I haven't touched my profile picture in years. Is this a bug?  Update: It appears my profile pic across all communities has been set back to 'Identicon'. 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It is now asking for my passport number and expiry date. That all seems reasonable. However, when I enter that I am a Canadian resident it asks for:  1) Street or hotel in US 2) City in US 3) State in US    My trip to DFW is only for a connecting flight. The fields are required so I can't submit my passport information without filling this in.    Is there something I'm missing? How to install Skype 4.3? I mistakenly typed cd // instead of cd /. To my surprise, current directory showed up as //.   What is that directory? Why does it exist?  apple@snipped $ pwd /home/apple apple@snipped $ cd / apple@snipped $ pwd / apple@snipped $ cd // apple@snipped $ pwd // apple@snipped $ cd /// apple@snipped $ pwd / How to make ttf-mscorefonts-installer package download fonts, after it says it installed? When I install "Installer for Microsoft TrueType core fonts" (ttf-mscorefonts-installer) from the Software Center on 14.04, it gives a positive response--but after this I get a download error. It seems that the files actually providing the fonts are never retrieved. Is there any way to resolve this? What are the assumptions for applying a quantile regression model? The question has been asked (one time) on CV before, but the answer is really imprecise and does not really answer the question in my opinion.  So: What are the assumptions for estimating a linear regression model via quantile regression?  To my understanding (and as several CV users have mentioned), quantile regression does not assume any specific distribution of the error terms - does that mean that, in a time series model, autocorrelation and heteroscedasticity do not have to be accounted for?  What about the other Gauss-Markov assumptions? I would assume that the assumption of no perfect multicollinearity has to be met when applying quantile regression, but do the parameters have to be linear? The linearity assumption only has to hold for the specific quantile I would assume.  Anyways - I do not find any backup for any of my thoughts in the scientific literature and I would appreciate a comprehensive answer. Thank you! In $ \triangle ABC$ show that $ 1 \lt \cos A + \cos B + \cos C \le \frac 32$ According to preliminary estimates, the project should be completed by 2020 at a cost of about RUB 40 bn.    or      According to preliminary estimates, the project should be completed by 2020 at the cost of about RUB 40 bn.    Are both options okay, or is the first better?   We are introducing the sum to the reader, so logically it should be the first option. But there's a lot of google-hits for "at the cost of" too. Beam size and material for 15 feet long swing I need to create a 15 feet span beam for kid swing (300 lbs load / 150 lbs and 2 swings assumption). One end will bolt to the tree (just for convenience).  What "treated lumber" lumber size and material I should use? Or is it better / cheaper to use metal pipe (from the plumbing department)?  I guess 4x8x16 or two (2) 2x8x16 would do it according to this website . But I do like a clean look of metal. My wife would concern about big piece of lumber running on top of the kid (that looks heavy visually). What is the equivalent of Persian idiom "When the reed blooms"? In Persian, we say "When the  blooms" when we want to  express that an event:   Never happens. (This is only the opinion of the speaker so it's not a fact) It's very unlikely to happen. It's going to take so long to actually happen. (It's so long that you'd be better off giving up expecting it)  3.1. We also use it when someone is always postponing something they have to do so you never actually see it happen.   For example:      Kyle: When are you going to pay my money back?      Cartman: When the  blooms.   Now I don't know if I got the plant's name right because there are some other versions like:        In any event, they have mentioned this plant in the idiom because apparently some versions of it actually never bloom. Others believe it's because farmers harvest them before they get the chance to bloom (e.g to make music instruments), so it never happens. I'd love to know the equivalent idioms or expressions for this.  PS: I only know "When pigs fly" but I'm not a fan of it because the word pig sounds a bit rude to me and partly because it implies a strong impossibility which doesn't work for the #3 use. If $ a,b,c\in \left(0,\frac{\pi}{2}\right)\;,$ Then prove that $\frac{\sin (a+b+c)}{\sin a+\sin b+\sin c}<1$ If $\displaystyle a,b,c\in \left(0,\frac{\pi}{2}\right)\;,$ Then prove that $\displaystyle \frac{\sin (a+b+c)}{\sin a+\sin b+\sin c}&lt;1$   $\bf{My\; Try::}$ Using $$\sin(a+\underbrace{b+c}) = \sin a\cdot \cos (b+c)+\cos a\cdot \sin (b+c)$$  $$ = \sin a\cdot (\cos b\cos c-\sin b\sin c)+\cos a(\sin b\cos c+\cos b\sin c)$$  $$ = \sin a\cos b\cos c-\sin a\sin b\sin c+\cos a \sin b\cos c+\cos a\cos b\sin c$$  Now how can i solve it after  that , Help required, Thanks Use of 'const' for function parameters How far do you go with const?  Do you just make functions const when necessary or do you go the whole hog and use it everywhere?  For example, imagine a simple mutator that takes a single boolean parameter:  void SetValue(const bool b) { my_val_ = b; }   Is that const actually useful?  Personally I opt to use it extensively, including parameters, but in this case I wonder if it's worthwhile?  I was also surprised to learn that you can omit const from parameters in a function declaration but can include it in the function definition, e.g.:  .h file  void func(int n, long l);   .cpp file  void func(const int n, const long l)   Is there a reason for this? It seems a little unusual to me. Is there a way to change the process group of a running process? Is there a way to change PID, PPID, SID of a running process? It would make sense for the answer to be no, but I'd like to make sure. WPF ComboBox without drop-down button Which characters unlock each scene?