Let $a&gt;0$ and $f:\Bbb R\to\Bbb R$ a function such that,$\forall x,y\in\Bbb R,\space 0&lt;y-x&lt;a\implies f(x)&lt;f(y)$. Does $f$ have to be injective or surjective?   This is what I did: Let $\alpha=y-x,\space\alpha\in(0,a)$. Then $f(x)&lt;f(y)\iff f(x)&lt;f(x+\alpha)$ and this holds for any $x\in\Bbb R$ because we can shift the interval $(0,a)$ to any $(x,x+a)$ on the $x$ axis. Therefore, I think, $f$ has to be strictly increasing, so it is injective, i. e., $\forall x,y\in\Bbb R$,$x&lt;y\implies f(x)&lt;f(y)$.  I also think $f$ does not have to be surjective and such counterexample is an exponential function $b^x,b&gt;1$, bounded below by $0$. Is my argument valid and if it is, should I improve anything?
Let $a&gt;0$ and $f: \mathbb R \rightarrow \mathbb R$ with a property $\forall x,y \in \mathbb R;$ such that $0\;&lt;\;y\;-\;x\;&lt;\;a$ $$f(x)\;&lt;f(y).$$ Prove if the statements are true or false:  $1)$ $f$ must be injective.  $2)$ $f$ must be surjective.  I tried to use a quarter of a circle with radius $r=a$ or a right-angled triangle and translate it. When I moved it, I saw new values of $f$. I concluded the maximum value in the previous interval can't appear again in another interval when the distance between one point in the first interval and one point in the second is less than $a$. I also considered using an absolute value function, but I haven't done anything. I think, but I haven't proven it, that the function must be bijective.
How to wrap lines correctly inside algorithmic I am having problems displaying algorithms. The wrapping is really ugly and I can't, for the life of me, find a fix that doesn't involve manually tweaking the hell out of everything, to the point that I might as well scrap the crappy algorithm packages and write everything out in TeX.  Is there any way to properly indent the following  \documentclass{article} \usepackage{algorithm} \usepackage{algpseudocode} \begin{document} \begin{algorithm}    \caption{My pseudo code.} \begin{algorithmic}[1]    \While{this line is sooooooooooo long and boring and too much for algorithmic to handle}       \State look at this state, this state is just too long for algorithmic to display it properly        \While{again this line is sooooooooooo long and boring and too much for algorithmic too handle}          \State look at this state again, this state is just too long for algorithmic to handle, I'm just going to switch to Word       \EndWhile    \EndWhile \end{algorithmic} \end{algorithm} \end{document}​     so that the output is similar to If $g\circ f$ is injective and $f$ is surjective then $g$ is injective 1N5819-E3/73  1N5819-E3/53 1N5819-E3/54   the first one is 4x cheaper than the last, why ?  aren't they all the same ? Cannot remove block comment in Eclipse after formatting If the magnetic field doesn't polarize does it follow the electric field path of propagation? or does it vanish? some \newcommand instructions not working Can a possessive S be attached to nouns that are not human beings (or animals)? Can a possessive S be attached to nouns that are not human beings (or animals)?  For example, instead of saying "the back of the chair", can I say "the chair's back"?   I remember learning that not everything can get a possessive s, but this issue is not clear to me. Why is $ 561 = 3*11*17 $ the smallest Carmichael number? A composite integer n is a Carmichael number if the only Fermat   witnesses for $n$ are those $a \in \mathbb Z_n^+$ which are not   coprime with $n$. The smallest example of such a number is $561 = 3 ·  11 · 17$   I think I understand the definition, but I don't quite understand why is the smallest Carmichael number is the above mentioned number. Please clarify. Why separation of variables works in PDEs? I have a script that needs to do some stuff based on file creation &amp; modification dates but has to run on Linux &amp; Windows.  What's the best cross-platform way to get file creation &amp; modification date/times in Python? Why does nothing show up in the Compositing tab? I am trying to follow Blender Guru's beginner tutorial. He went to the Compositor tab and was using it to Denoise an image but when I got there, no nodes show up. Can you guys help? Displaying and/or searching across thousands of contacts? The Need  Our application has a chat feature that allows our users to chat with other users within their database.   However, we're adding a feature that allows clients to share data between multiple databases as an "Enterprise" configuration, and we've been asked to update the design of the chat feature to allow users to chat with other users within any of those databases.   We've looked at the numbers of possible users in these scenarios, and the number of chat users in these shared database environments can be as many as 5,000.   Users need to be able find people they need to chat with out of these thousands of contacts. However, up til now, we've had a list of contacts the users can browse. Since that is a user expectation, we were hoping to still maintain a browse-able list of contacts. We have ruled out trying to display a list of 5,000 contacts, as this doesn't seem to be anywhere close to useful or usable.   We've come up with three approaches to solve the problem:  Option One  Show a list of contacts from the local database that user's can browse. Have a second tab that allows users to search the "Full Directory", returning results from multiple databases.     Pros:   Allows us to show a browse-able list of relevant contacts  Clearly indicates to users that another source of chat contacts are available   Cons:   Search would be limited to the current tab, meaning users would have to be in the right tab to return the right result Adds a possible source of user confusion: whats the difference between "contacts" and the "full directory"? Adds another tab to what should be a simple chat list     Option Two  Show a list of contacts from the user's local database, but if they search for a contact, include results from other databases as well.     Pros:   Maintains a list of contacts the user can browse Avoids adding adding another tab One search field searches all sources   Cons:   Searching a finite contact list and returning results that weren't in the contact list may defy user's mental model of how the search will work. Not clear that another source of contacts exist. May need to add explanatory text that says "Search to see contacts from the full directory".      Option Three  Add a dropdown that allows a user to select a what to search in: the local database, or the full directory.     Pros:    No additional tab required Indicates to users that another source of chat contacts are available, though not as clearly as option 1   Cons:    Search still isn't universal: the right search category must be selected to return the right result Users may not notice the dropdown or understand intuitively what its function is. May require explanatory text in search results, reminding the user they can expand the search to the full directory     Thoughts?  Right now I'm leaning towards option two, but is there another option I'm not considering here? Is there another alternative worth considering?   Thanks! Oneboxing a onebox from a different chat server generates a weird URL So if you try to onebox a onebox from a different chat server, . Basically, it generates this URL:    Now, obviously that isn't right. I've also tested this , and I'm assuming it's broken on chat.SO too. Can this be fixed? MacBook Pro 2020 Fan Issues I recently purchased a MacBook Pro 2020 13&quot; (1.4 GHz, 8GB Ram, i5). For some reason, when I open Chrome and watch Twitch my computers fans turn on and go crazy. After opening Activity Monitor, I noticed that Google Chrome Helper (Renderer) uses around 50-80% of my CPU. Also, sometimes when I run code on my MacBook Pro, the fans turn on when the program is compiling/running but slow down a few minutes after terminating the program. Also, when watching Twice on Safari, the fans don't come on. However, around 50-60% of the CPU is still being used. Should I be concerned about the CPU usage (I am nervous because I only recently purchased the laptop)? Why is the de-facto standard sigmoid function, $\frac{1}{1+e^{-x}}$, so popular in (non-deep) neural-networks and logistic regression?   Why don't we use many of the other derivable functions, with faster computation time or slower decay (so vanishing gradient occurs less). Few examples are . One of my favorites with slow decay and fast calculation is $\frac{x}{1+|x|}$.  EDIT  The question is different to  as I'm only interested in the 'why' and only for the sigmoid. I used an SRTM DEM to create a hillshade in QGIS 2.2.0. However, the resultant hillshade seems to be vertically exaggerated. Also, changing the settings for scale and z-factor does not have any effect on the resultant hillshade. I have tried both options in QGIS, Analysis-> DEM (Terrain models) and Terrain analysis -> Hillshade. Both methods gave the same results.  Is this an isolated incident, a bug, or am I missing something? Is it possible to get Inkscape's bucket fill to entirely fill regions marked by lines? I've noticed there's a slight gap between shapes and lines in which bucket-fill is applied (to the enclosed areas found therein) and the colour fill in Inkscape, is there any way of removing this gap (besides the older-fashion manual way by manipulating the borders of the bucket fill, which is very tedious)? I am using Eclipse 3.3 ("Europa"). Periodically, Eclipse takes an inordinately long time (perhaps forever) to start up. The only thing I can see in the Eclipse log is:       !ENTRY org.eclipse.core.resources 2 10035 2008-10-16 09:47:34.801     !MESSAGE The workspace exited with unsaved changes in the previous session; refreshing workspace to recover changes.   Googling reveals  that I remove the folder:  workspace\.metadata\.plugins\org.eclipse.core.resources\.root\.indexes   This does not appear to have helped.  Short of starting with a new workspace (something which I am not keen to do, as it takes me hours to set up all my projects again properly), is there a way to make Eclipse start up properly? The Ubuntu installer does not allow me to install on NTFS partitions, but certain circumstances requires me to do so. Is it possible? Ways to give users some specific education about question quality and topicality Important: This question is being asked here, and not on the Programmers Meta,  because I believe that several SE sites suffer from this same, specific problem.    On Programmers, I see a pattern emerging.   New User sees the "Programmers" site title, and asks their incomplete, "fix my broken code" question. Community members comment that their question is off-topic, and recommend Stack Overflow.  New User says "thanks" and leaves, leaving their undeleted question for the community to clean up.    What seems clear to me is that:   New User has never seen the "What kinds of questions can I ask about here" page, or if they have, they've clicked through it without reading it. New User doesn't know how to move their question to the right place properly  (i.e. post on new site, delete from old site). Community cleanup of such questions is onerous.  The moderators on Programmers, following the principle that the community should moderate itself for the most part, are not proactive about removing such questions unless they're especially egregious. Migration is even more onerous.   OK, so we have the .  Here's what it says about topicality, about halfway down the page:    Note to those who are confused about Programmers' site scope: it's all right there, in black and white.  However, I do notice a problem.  It's circled in red.  That's our fault, the fault of the Programmers community.  What we really meant to say was "Don't ask your code troubleshooting questions here; those belong on Stack Overflow."  Coding tools deserves its own bullet.  We should fix that.  However, what are the chances that the user is actually seeing this, and evaluating whether or not they should ask their question based on this?  So here's my question, in two acts:  ACT I:  Is there a way to highlight the pain points of a particular site to new users specifically, so that we can be very clear that we don't want those questions that are clearly and unambiguously off-topic on our site, before they ask their question?  ACT II: Failing that, is there a way that we can fast-track the removal of such questions so that they no longer pollute our front page?