Definition of continuity of open balls or delta I know how to prove the theorem of the following:  A function f is continuous if and only if the preimage of any open set in Y is open in X.  I am wondering how to show that the theorem is equivalent to the two different definitions of continuity of i) open balls and ii) delta epsilon.
The equivalence of open-set definition of continuous mappings to the $\epsilon-\delta$ definition A function $f: M\rightarrow N$ is defined to be continuous iff $\forall$ open set $U\subseteq N$, the preimage of $U$ is also open.  I was trying to prove that such a definition would be equivalent to the $\epsilon -\delta$ definition in a metric space, and I could only do so if I assume $f$ is surjective, or otherwise a counterexample would be $g: \mathbb{R}\rightarrow\mathbb{R}$ where $g(x)=x^2$. As if I take $U=(-1,1)$, then its preimage is $[0,1)$, which is neither open or closed. In fact, any $U$ containing elements that cannot be achieved by $g$ would face the same problem.  So I was wondering that have I remembered the definition of preimage wrong, or do I need to make sure that $U$ must be chosen from the image set of $f$?
US tourism/business visa from non-resident country What is the pronunciation of the possessive words that already end in s? Possible Duplicate:           My name is Greg — this is Greg 's post.  If my name ended with an 's', I am aware of the proper apostrophe usage (James&nbsp;→ James'), but how should this be pronounced?  Phonetically, am I Jameses best friend or James best friend? The RSA signature operation is basically the same as encrypting with the private key.  In particular, both operations use the same kind of keys.  Is it safe to use the same RSA keypair both for encryption / decryption and for signing / verification? My online friend is asking for money in order to visit my home country. Is this a legit request or a scam? Git for beginners: The definitive practical guide How do I add a DNS server via resolv.conf? Is /etc/resolv.conf useless in  (Precise Pangolin)?  I see that the DNS server information is stored in  now. The nmcli command line tool can list that for you.  If I want to add one more  server, will adding it to /etc/resolv.conf by using the resolvconf package help? how to change the distance between tick mark and tick label Direction of friction for object that rolls with slipping Currently I am using Outlook 2010, and sometimes customers claim that they will received an attachment in Windows.dat format.  I have searched online for possible solutions, but mostly the suggestion is to change the sending format to plain text. However, by using plain text format, there is some limitation.  So I just wonder, besides the plain text format, is there a ways to solve the winmail.dat issue?  Besides, does plain text format really 100% avoid the issue? If a function $f$ defined on $[a,b]$ and differentiable everywhere, does it mean that its derivative $f'(x)$ is continuous everywhere on $[a,b]$?  My understanding is 'yes'. Because if there were 'a gap' in $f'(x)$, we could integrate back to $f(x)$ and show that where the gap is, there would be a sharp "angle" where $f(x)$ is not differentiable. Confused! Are there any differences between "I have to go", "I had to go", "I've had to go", "I get to go", "I got to go", "I've got to go"? I am Confused!   Are there any differences between "I have to go", "I had to go", "I've had to go", "I get to go", "I got to go", "I've got to go"?  In which situation is each term used?  I checked this dictionary , &amp; found that      get to do something:  to have the opportunity to do something   Seem like if you have a opportunity to do something, then use "get".  The expression is in present but why we use present perfect tense (Ex: I've got to go) here? Making SSD & HDD work together like a hybrid Stargate - Destiny's Gate Address from Icarus base Simple question - What is the address they dial to connect with the Ancient ship Destiny in Stargate Universe from Icarus base.  I have found the address on  - however they have an unconfirmed order, and not sure of the validity of this.  Any help deciphering the above would be great! Wording of "possible/exact duplicates" when closing the question When marking question for "duplicate", the reason in dialog window is "exact duplicate", yet when the process is completed there's a comment added that says "possible duplicate". Maybe those two should be evened out to either "possible duplicate" or "exact duplicate"?  Edit: Take  for instance, it says "Possible" on top but "exact" at the bottom    and the top remark was added by the system only after the question was closed, i.e. after it was concluded that it was an exact duplicate rather than a possible one.  So the question is: can the text on top be changed to "Exact Duplicate" please? I have an existing windows application, for which I have decided to create a Mac version. The problem is that the current UI involves a lot of buttons, tabbed panels, multiple windows etc.  This UI had been rated as intuitive and easy to use and straightforward  by many customers and editors. But a beginner (very less experience) may find in confusing a little, but it has been complemented with tutorial.      When creating the Macintosh variant, should I focus on creating a simplified and minimalistic UI (this does not mean the current UI is complex, but I think Mac user's prefer a simple UI right?) or should I try to copy the exact same thing?  By reducing the UI some of the features may be missing. But it can be added later if needed. Should I focus on a fresh approach (this is easier rather than copying the complex UI pattern) or should I go with the old recognized setup? When reading about  on the OALD, I saw the following note:     The difference between someone and anyone is the same as the difference between  and . Look at the notes there.   The note that I read for some is the following:     In negative sentences and questions any is usually used instead of some:           I don't want any more vegetables.     Is there any wine left?           However, some is used in questions that expect a positive reply:           Would you like some milk in your coffee?     Didn't you borrow some books of mine?      Does that apply to someone too? Should I use someone in a question when I am expecting a positive answer?      Does someone know the answer? How to assign a SSL Certificate to IIS7 Site from Command Prompt Can you advise me whether it is possible or not to assign a SSL Certificate to a website in IIS7 using the APPCMD application?  I am familiar with the command to set the HTTPS Binding  appcmd set site /site.name:"A Site" /+bindings.[protocol='https',bindingInformation='*:443:www.mysite.com']   and how to obtain current mappings   %windir%\system32\inetsrv\Appcmd   but can not seem to find any way to map a site to a certificate (say the certificates hash for example) When do I use the PHP constant "PHP_EOL"? When is it a good idea to use ?  I sometimes see this in code samples of PHP. Does this handle DOS/Mac/Unix endline issues? Getting hybrid graphics to work nvidia-prime GT650M I recently made the switch to Ubuntu. Everything on my laptop works fine except my hybrid graphics card. It is Intel graphics combined with Nvidia GeForce GT650M. I tried Bumblebee, but primusrun doesn't work (segmentation fault, core dumped) and optirun works with glxspheres64, but steam games look worse with optirun than with Intel graphics (e.g. Surgeon Simulator 2013 runs fine with Intel and hangs a lot with optirun)   I tried Nvidia Prime, but it gives me a black screen on startup after install, so I can't do anything and have to power off my PC with the power button. I also tried nvidia-prime 0.5 on Ubuntu 14.04, and then I get 'low-graphics mode' and I can't get to Unity. There are many different kind of options that you can do  in order to start using LaTeX on Ubuntu. I would prefer doing that through Eclipse, but if there is a better way of doing it, feel free to write your opinion on why using this instead of something else.