Given: $f: X → Y, x\in X$, and $f$ is continuous at $x$, $f$ is continuous on $X$.  Prove that $f$ is continuous on $X$ if and only if for each open subset $V$ of $Y$, $f^{-1}(V)$ is an open subset of $X$.
Let $ X \subset \mathbb{R}$ be a non-empty, open set and let $f: X \rightarrow \mathbb{R}$ be a continuous function. Show that the inverse image of an open set is open under f, i.e. show:  If $M \subset \mathbb{R}$ is open, then $f^{-1}(M)$ is open as well.  I think that this should basically follow from the definitions, but I'm still having some troubles with the proof.  Suppose $M \subset \mathbb{R}$ is open, then it follows that $\forall z \in M, \exists r &gt; 0: K_r(z) \subseteq M$  Since f is continuous it follows for each $x_0 \in X$ that for each $\epsilon &gt; 0 ,\exists \delta &gt; 0 : |f(x)-f(x_0)|&lt; \epsilon, \forall x \in X: |x-x_0|&lt;\delta$  Now I need to show that $f^{-1}(M)$ is open, so that for each $z \in M, \exists r&gt;0: K_r(f^{-1}(z_0)) \subseteq f^{-1}(M)$  I'm not sure how to prove this. I guess since $z = f(x)$ and since M is open we have that $\exists r&gt;0:\{z:|z-z_0|&lt;r\}\subseteq M$. And since f is continuous: $|f(x)-f(x_0)| = |z-z_0|&lt; \epsilon$ for all $|x-x_0|&lt;\delta$, but since $|x-x_0|=|f^{-1}(z)-f^{-1}(z_0)|&lt;\delta$ it somehow follows from that that $f^{-1}(M)$ is open?!  Which is just horrible and probably completely wrong. But I'm really confused from all the definitions right now and need some help please.
If a person holds prejudice against people because of their nationality, would that be considered racist? Would it be considered racist if a store owner believes all Canadians are thieves and does not let any Canadians into his store? Racism pertains to discriminating based on race, and (correct me if I'm wrong) Canadians are not a race. What would be an appropriate word to describe an individual such as the  store owner?  The discrimination is specific to Canadians, not foreigners in general. I'm looking for a word that could be applied to someone who lives in the country they discriminate against. For example if a Canadian announces "all Canadian culture is stupid!" what would that be called? Why can I see the output of background processes? For example, when I type ping 8.8.8.8 &amp;, why can we see ping process working? And when I type find / -name '*test*' &amp; , it's also seen on monitor.  Why is that? Is it not really in background? How to prevent the screen lock while watching video (Netflix, Youtube, Amazon Prime) in Ubuntu 17.10 In earlier Ubuntu versions like 16.04 and 17.04 while watching YouTube videos in full-screen mode, the screen would not get locked automatically.  In Ubuntu 17.10 while watching videos in browser, system is getting locked. So Ubuntu 17.10 is considering this scenario as user is not doing anything.  Kindly suggest a solution for that. Stack Overflow, Stack Exchange, Area 51: What's the difference in a nutshell? Stack Overflow, Stack Exchange, Area 51: What's the difference in a nutshell? I've read the faq's and they all seem to be the same to me - Just Q &amp; A on a variety of topics.  What's the difference in a nutshell? If I have a software engineering question, which site is best for me to use? Comma — a punctuation error? Consider:     This book provides a solid psychological understanding of these experiments, and adds a few expansions and conclusions of its own. It also provides …   My question regards the comma. I have been told that having the comma there is an error.  According to  this seems to be the case according to “some style guides”. The reason is that the stuff after the comma is not independent; it’s dependent on what comes before.  What do you think? It would seem to me the comma helps a lot, because those two “and”s without any comma make the sentence rather cluttered. That’s why I put the comma there. So I’m not sure. Change pivot or local origin of an object How can I change the local origin of an object, without changing the object location in the world space.  This is the object located at position (0,0,0) in the world space    This is the same object located at position (0,1,0) in the world space, but with its local origin offset so that the object is in the same place    Is there a way to do this in a single step? What's the bad magic number error? A set with a finite integral of measure zero? Prove, or give a counter example: Let $\mu$ be a finite positive borel measure on $\mathbb{R}$. Then $\int (x-y)^{-2} d \mu (y) = \infty $ almost everywhere on $\mu$ (for the selection of x's).  This is a question I had in an exam, and the answer is supposed to be presented in less than 30 words, so there must be something quite simple I'm missing. Why shouldn't we use words such as 'here' and 'this' in textlinks? Over the years of surfing on the web I have found numerous examples of people saying that you shouldn't use the word "here" in a textlink etc. Personally I like the idea of using the word, because it is so straightforward for your visitors to know where to click.  The only reason that I can truly think of is for SEO purposes. The word here doesn't have much value as a keyword as such.  I would like to know some other opinions on why we should/shouldn't use word like "this" and "here" in a textlink. My only problem is making them line up three-across and have equal spacing. Apparently, spans can not have width and divs (and spans with display:block) don't appear horizontally next to each other. Suggestions?  &lt;div style='width:30%; text-align:center; float:left; clear:both;'&gt; Is what I have now. Are the hoses in toilets in Asia something I should know about? I have noticed random bits of hose without proper fittings before I think for example in India and just assumed they are for use by the cleaners or for filling the water jug in .  But Now I'm travelling in Malaysia and Singapore and there are more western toilets than squat toilets in the places I've been yet proper hoses with proper fittings are very common in toilets:  (Hose next to a Western style toilet in my hostel in Chinatown, Kuala Lumpur.)  At one place in Singapore there were even signs on each cubicle door stating whether it contained a western or squat toilet, but also whether there was a hose! (I didn't take a photo of the sign, unfortunately)  This makes me think there is a use for the hose for clients, not just cleaners. But for what purpose? Should I know how to use it as a tourist? Please tell me it's not a kind of alternative to a bidet!  Possibly related is the fact that many toilet floors are awash with water. This can be a big problem if you're not wearing waterproof shoes or have to lower your trousers and keep them dry. Other than this toilets are pretty clean and I've seen both the hoses and the wetness in what I would regard as somewhat "fancy" places. Can a class overload methods that also exist in the publicly inherited interface? It seems like this is unambiguous and useful, but compilers (VC, Intel, GCC) all complain, at least by my construction. Below is a toy example. The inherited rebound() function has two clear overloads, yet this will not compile.  If you rename the rebound() method in either class, it works fine, but if they share the same member function name (even though they're overloaded with different argument types!) you get a fatal error of "too few arguments to function call."  The workaround is trivial (I'll just rename the methods) but I'm just trying to understand if this is a C++ restriction (and why it would be).   #include  class Bound { public:   Bound() : x0(0.0), x1(0.0) {};   Bound(double x) : x0(x), x1(x) {};   double width() const {return x1-x0;}   void rebound(const Bound *a, const Bound *b); private:   double x0, x1; };  void Bound::rebound(const Bound *a, const Bound *b) {   if (a && b) {     x0=std::min(a->x0, b->x0);     x1=std::max(a->x1, b->x1);   } }  class Node : public Bound { public:   Node(double x) : Bound(x), left(0), right(0) {};   Node(Node *a, Node *b) : left(a), right(b) {rebound();}   void rebound() { rebound(left, right); } private:   Node *left;   Node *right; };   int main() {   Node A(1.0);   Node B(2.0);   Node C(&A, &B); } Pop up option are not popping up after snipping the area How to create local total counters? I would like to write page numbers like this '7/9' in all minted environments of my document. The first number ('7') is the current page and the second number ('9') is the maximum number of page breaks the source file has been split into.  I was planning to use the package totcount but I realized that it was only considering the last value of the counter at the end of the document and it couldn't manage counters locally to an environment.  Here is a totally stupid example, but it illustrate the problem:  \documentclass{report}  \usepackage{totcount} \newtotcounter{sectnum}  \let\oldsection\section \renewcommand{\section}{%   \setcounter{sectnum}{0}%   \regtotcounter{sectnum}%   \oldsection% }  \let\oldsubsection\subsection \renewcommand{\subsection}{%   \stepcounter{sectnum}%   \oldsubsection% }  \begin{document}  \section*{One}  \subsection*{SectionOne} \thesectnum / \total{sectnum}  \subsection*{SectionTwo} \thesectnum / \total{sectnum}  \subsection*{SectionThree} \thesectnum / \total{sectnum}  \subsection*{SectionFour} \thesectnum / \total{sectnum}   \section*{Two}  \subsection*{SectionOne} \thesectnum / \total{sectnum}  \subsection*{SectionTwo} \thesectnum / \total{sectnum}  \end{document}   Gives:    Where we would have liked to have: '1/4', '2/4', '3/4', '4/4' and '1/2', '2/2'.  So, is there a way to get this kind of feature ? Why does HMAC use two different keys? Suppose $H$ is a hash function; why is $$H(k\mathbin\|H(k\mathbin\|m))$$ not secure?  See this . In there, indeed two keys are used and the mac algorithm is $$H(k_1\mathbin\|H(k_2\mathbin\|m)).$$ Why don't we use $$H(k\mathbin\|H(k\mathbin\|m)),$$ which has only one key? Neatest proof that set of finite subsets is countable? I am looking for a beautiful way of showing the following basic result in elementary set theory:     If $A$ is a countable set then the set of finite subsets of $A$ is   countable.   I proved it as follows but my proof is somewhat fugly so I was wondering if there is a neat way of showing it:  Let $|A| \le \aleph_0$. If $A$ is finite then $P(A)$ is finite and hence countable. If $|A| = \aleph_0$ then there is a bijection $A \to \omega$ so that we may assume that we are talking about finite subsets of $\omega$ from now on. Define a map $\varphi: [A]^{&lt;\aleph_0} \to (0,1) \cap \mathbb Q$ as $B \mapsto \sum_{n \in \omega} \frac{\chi_B (n)}{2^n}$. Then $\varphi$ is injective hence the claim follows. (The proof of which is what is the core-fugly part of the proof and I omit it.). I've just received a memo which says (effectively)      As more people leave, there will be less people available.   I want that word to be fewer. Are there guidelines for which word ought to be used when? Sudoers blacklist How to remove the "3D Objects" entry from Windows 10 Explorer? The 2017 Fall Creators Update to Windows 10 added a "3D Objects" folder to the "This PC" view of Windows Explorer:    How can this be removed? Setting up LVM Snapshot as a backup/restore point in ubuntu Setting up LVM Snapshot as a backup/restore point in ubuntu. Why? Well, as I see it, with all the great usefulness of Linux, its biggest drawback is with its mixed up way of installing apps. All the dlls go to one directory. All the configuration files to another. And the binaries are also put in a bunch. Yes, Windows works the same way more or less, but with the nature of system/server software installed on Linux, being that most of the software installed is only about 95% mature at any given time, messing up your system twice a month is very probable.. Of course, we can backup the system before and after each installation, but that takes a million years. System backups (as oppose to just data/files backup) requires you to turn off you computer and use some form of bootable live CD and a backup media. And, if the drive is more than 15% full, it might take a long time to do the backup, even with modern hardware. So, how can this be done?