How to find all and every continuous function $ f $ from $ \mathbb{R} $ in $ \mathbb{R} $ that verifies $$ f(x+y)=f(x)\cdot f(y)\quad\forall x,y\in\mathbb{R} $$  All I could find is $ f(x)=f\left(\frac{1}{-x}\right) $... but I don't like that answer at all... any hints please??
I was wondering if there is a name for a function that satisfies the conditions  $f:\mathbb{R} \to \mathbb{R}$ and $f(x+y)=f(x) \cdot f(y)$?  Thanks and regards!
Compute the limit of $\sqrt{1-a}\sum\limits_{n=0}^{+\infty} a^{n^2}$ when $a\to1^-$ I need some suggestions, hints for the limit when $a \to 1^{-}$ of $$\sqrt{\,1 - a\,}\,\sum_{n = 0}^{\infty}a^{n^{2}}.$$ Unsure whether to use "enquire" or "inquire" How to add a PPA on a server? The server version of Ubuntu does not have the  command. How can I add a  (PPA) to the server without this command? How do you effectively study physics? How does one read a physics book instead or just staring at it for hours?   (Apologies in advance if the question is ill-posed or too subjective in its current form to meet the requirements of the FAQ; I'd certainly appreciate any suggestions for its modification if need be.) Verbatim environment that can break long lines? I want to generate a PDF of user-submitted text using LaTeX. To handle crazy user input, I first thought about using the verbatim package, but of course it doesn't break up too long lines. Is there some package that works similar to verbatim (i.e., accept any input) but formats the text nicely? Install App Image such as beaker browser and pin it to dock (favorites) in Ubuntu 20.04? A simple pendulum suspended from the ceiling of a stationary lift has period $T_0$. When the lift descends at steady speed, the period is $T_1$. When the lift descends with constant downward acceleration, the period is $T_2$. Explain why $T_0 = T_1 &lt; T_2$.  I know that for small angles of oscillations, $T \approx 2\pi\sqrt\frac{l}{g}$ where $l$ is the length of the pendulum. For the first two cases, since $\sum F = 0$, the value of $g$ is apparently the same. However, I am unable to explain why the period is largest for $T_2$. If $g$ is the same for all cases, shouldn't their periods be the same? Efficiently replace all accented characters in a string? For a poor man's implementation of near-collation-correct sorting on the client side I need a JavaScript function that does efficient single character replacement in a string. Here is what I mean (note that this applies to German text, other languages sort differently):  native sorting gets it wrong: a b c o u z ä ö ü collation-correct would be:   a ä b c o ö u ü z  Basically, I need all occurrences of &quot;ä&quot; of a given string replaced with &quot;a&quot; (and so on). This way the result of native sorting would be very close to what a user would expect (or what a database would return). Other languages have facilities to do just that: , in , , . But what about JavaScript? Here is what I have right now. // s would be a rather short string (something like  // 200 characters at max, most of the time much less) function makeSortString(s) {   var translate = {     &quot;ä&quot;: &quot;a&quot;, &quot;ö&quot;: &quot;o&quot;, &quot;ü&quot;: &quot;u&quot;,     &quot;Ä&quot;: &quot;A&quot;, &quot;Ö&quot;: &quot;O&quot;, &quot;Ü&quot;: &quot;U&quot;   // probably more to come   };   var translate_re = /[öäüÖÄÜ]/g;   return ( s.replace(translate_re, function(match) {      return translate[match];    }) ); }  For starters, I don't like the fact that the regex is rebuilt every time I call the function. I guess a closure can help in this regard, but I don't seem to get the hang of it for some reason. Can someone think of something more efficient?  Answers below fall in two categories:  String replacement functions of varying degrees of completeness and efficiency (what I was originally asking about) A  of  , which is now  among JS engines (not so much at the time of the question) and could solve this category of problem much more elegantly. If I'm given a complex number (say $9 + 4i$), how do I calculate its square root? I have these two CREATE TABLE statements:   CREATE TABLE GUEST (   id int(15) not null auto_increment PRIMARY KEY,   GuestName char(25) not null );  CREATE TABLE PAYMENT (   id int(15) not null auto_increment   Foreign Key(id) references GUEST(id),   BillNr int(15) not null );   What is the problem in the second statement? It did not create a new table. How can I add a new user as sudoer using the command line? After I add a user using adduser, I can't see it via System > Administration > Users and Groups unless I log out and then log in again. Is that normal?  Also, can I set a newly added user as a sudoer or do I have to change that only after adding it? How can I do that via the shell?  Finally, can I delete the original user that was created upon initial installation of Ubuntu, or is this user somehow 'special'? Is the "the ... the ..." construction governed by some grammatical rule? a) The more I study, the less I learn. b) More I study, less I learn.  Could we rewrite the sentence shown under the letter a) in the form shown under the letter b) without breaking any grammatical rule? If not, why not? Compile TeX (directly) into SVG using the command line Is there a way to directly compile TeX code into the SVG image format (rather than going the detour over TeX => PDF => SVG)?  I found this question:   But I don't want to do it online - it should be a command-line call. I followed  to create a soccer ball.  Now I want to texture it in a way that every pentagon has an image inside itself and the exagons are white, but I also want everything to be a little glossy. I managed to successfully UV Map the ball and paint the texture in photoshop but now I am having big problems in applying the texture material to the mesh.    I've created a new material in Blender Render mode with Diffuse and the texture applied to it. I also set the Texture mapping to UV. I've gone in edit mode, selected all the mesh and applied the material. Unfortunately in Texture View Mode I see just a white ball and while rendering I see everything's black.      If I try to do this in Cycles Render and use a Mix Shader with the FAC set to the texture image I get what I want in the Texture View Mode but while rendering I just see everything black and white of my colored images.        I hope I've been clear enough and that someone will help me in this easy/difficult task! Thank you in advance How the `\alignat` works in ams package I have read the related question(), which had detailed answer but still don't know what the meaning of the \alignat's argument. It is said in  that      This environment takes one argument, the number of “equation   columns”: count the maximum number of &amp;s in any row, add 1 and and   divide by 2.   What is the meaning of adding one in &amp; number and divide by two? I presumed the argument is given by the user, not automatically generated by the system? How to view and export database content? I'm trying to view the actual entries, not the backend identifiers.  &gt; db.&lt;collection&gt;.find({"_id": "486b4ca55b144b119f4551417c5798c8"}).pretty()  {     "_id" : "486b4ca55b144b119f4551417c5798c8",     "date" : ISODate("2017-04-20T05:47:06.176Z"),     "content" : ObjectId("58f84b5a0318762cb6e230a6"),     "digest" : "d064a874c872d75f6ab67c04d8b79b26f65f9b4f",     "short" : "5f9b4f",     "size" : 2374694 }   Also tried mongoexport:  $ mongoexport --host 127.0.0.1 --port &lt;port&gt; -o test --pretty --db &lt;database&gt; --collection &lt;collection&gt;   This also shows backend data. I just want to see what was uploaded to the database. The actual content itself. How to install Windows on GPT disk without UEFI BIOS? Add table row in jQuery Making all row sums and column sums non-negative by a sequence of moves Cartoon with a young unicorn and a demon In the late 80s or early 90s I remember watching an animated series that followed a young unicorn and a demon.   In one episode the unicorn gave the demon his horn, and the unicorn almost died.   In another episode there was a man who was able to turn people into human shaped building blocks and built a huge tower from them.   Those are the two scenes I remember with any clarity.   Does anyone have any idea?