Can anyone give me a classification of the real functions of one variable such that $f(x)f(y)=f(xy)$? I have searched the web, but I haven't found any text that discusses my question. Answers and/or references to answers would be appreciated.
Let  $f(xy) =f(x)f(y)$ for all $x,y\geq 0$. Show that $f(x) = x^p$ for some $p$.    I am not very experienced with proof. If we let $g(x)=\log (f(x))$ then this is the same as  $g(xy) = g(x) + g(y)$   I looked up the hint and it says let $g(x) = \log f(a^x) $   The wikipedia page for functional equations only states the form of the solutions without proof.  Attempt  Using the hint (which was like pulling a rabbit out of the hat)  Restricting the codomain $f:(0,+\infty)\rightarrow (0,+\infty)$ so that we can define the real function $g(x) =  \log f(a^x)$ and we have $$g(x+y) = g(x)+ g(y)$$   i.e $g(x) = xg(1)$ as $g(x)$ is continuous (assuming $f$ is).  Letting $\log_a f(a) = p$ we get $f(a^x) =a^p $. I do not have a rigorous argument but I think I can conclude that $f(x) = x^p$ (please fill any holes or unspecified assumptions) Different solutions are invited
I came across this strange code snippet which compiles fine:  class Car {     public:     int speed; };  int main() {     int Car::*pSpeed = &amp;Car::speed;     return 0; }   Why does C++ have this pointer to a non-static data member of a class? What is the use of this strange pointer in real code? Is there an implementable (non-theoretical) way to prove if email address belong to real email account? Is there any way other than instanceof operator for object type comparison in java? XML DML for 2008 R2..How to modify nested Elements Using all the letters of the word ARRANGEMENT how many different words using all letters at a time can be made such that both A, both E, both R both N occur together . Falsely accused witch befriends transforming dragon she was to be sacrificed to Story is in a medievalish setting and starts with a woman who has been falsely accused of being a witch being taken by 2 men to be sacrificed to a dragon.  She ends up befriending the (male) dragon, who is able to transform into various forms, including human, and can speak only the language of whatever form he is in. e.g. can speak English in human form, but can't in dragon form.  He needs to return to his dragon form every day at sunrise or he dies.  Contact with iron or steel causes him pain and prevents him from transforming.  Near the end of the book, he's been put in handcuffs by one of the two men from the start and the woman has to get the assistance of the other one (village blacksmith?) to remove them before sunrise, which they manage to do just in time.  I think I read it maybe 10 years ago.    I also remember it was a very short book, probably only about 100 pages. I want to know is there a way to render Shadows only in Blender's new EEVEE Render Engine. I want my objects in a separate collection and a ground plane on a separate collection(to receive shadows from the objects collection).I want to render the shadows only and not the objects. btrfs with multi devices fails automount in fstab I'm trying to automount btrfs  on 12.04-desktop-amd64. During bootsplash, fstab failed to mount btrfs and asked skip or manually prepare.  On the rootshell, btrfs device scan and mount -a could automount properly. It seems "btrfs device scan" was missing from bootstrap.  Where should I add the line to prepare bootstrap? I want to draw a TikZ matrix with its border and I would like to define the distance between most external matrix nodes and matrix border. Something like a matrix inner sep. If I change inner sep in matrix options declaration, all nodes inside the matrix also change its inner separation.      I know how to do it with fit library. I just want to know if its possible without it. Closed formula for the sums $\sum\limits_{1 \le i_1 < i_2 < \dots < i_k \le n} i_1 i_2 \cdots i_k $? I've worked out a few summation formulas, and I am hoping to find a pattern. Unless I have made a mistake somewhere, we have the following identities:  $$\sum_{1 \le i \le n} i = \frac{(n+1)n}{2} $$  $$\sum_{1 \le i &lt; j \le n} ij = \frac{(3n+2)(n+1) \, n \, ( n-1)}{24}$$   $$\sum_{1 \le i &lt; j &lt; k \le n} ijk = \frac{(n+1)^2 \, n^2 \, (n-1)(n-2) }{48}$$   It seems clear that $$p_k(n)= \sum_{1 \le i_1 &lt; i_2 &lt; \dots &lt; i_k \le n} i_1 i_2 \cdots i_k $$ is a polynomial in $n$ of degree $2k$. But is there a nice closed-form formula for it? Maybe in terms of its factorization?  Comment: I realize that what I am looking for is the coefficient of $t^k$ in the expansion $$(1+t)(1+2t) \dots (1+nt),$$ so maybe generating function techniques could be helpful. But the main way I know to extract the coefficient of $t^k$ is to take derivatives $k-1$ times and on the surface of things that looks like a huge mess. Notification email from stackoverflow contains "<3" after zip code I noticed "&lt;3" appears at the very end of the my stackoverflow notification emails (after the zip code). Is this problem related to gmail? Java: Load class from string Are any of these verb phrases grammatical?        allows the user of modeling and resolving     allows the user to model and resolve      Which version of the following sentence is correct/better?        In particular, it allows to acquire a signal using a sample rate significantly lower than the one dictated by the Nyquist criterion.   In particular, it allows acquiring a signal using a sample rate   significantly lower than the one dictated by the Nyquist criterion.      I searched Google Scholar for "" and "", and it seems people use both more or less with the same frequency. It seems that with some profiles the Gravatar image is broken. I am not sure if this is an issue with SE or Gravatar, or both, so posting here.  As an example,  shows an error on , which shows a HTTP 500 on Gravatar.  Profile image on SO:        Image on Gravatar:        Switching to the MSE profile of the user, the image is working (but has a different hash).  Is this a bug on the SE side? Can anything be done to make Gravatar fix this?    Update 2016-10-04:  It happened to me again two times today. The last user profile was . Calling the service two or three times fixes the issue. I want to construct a field with $125$ elements. My idea is to consider the polynomial ring $\Bbb F_5[x]$. It is enough to find an irreducible polynomial $f\in \Bbb F_5[x]$ of degree $3$ because then $\Bbb F_5[x]/(f)$ is a field with exactly $5^3=125$ elements.  How do I find an irreducible polynomial of degree $3$ in $\Bbb F_5[x]$? Do I need an airport transit visa through Switzerland if I have no Schengen visa? Am traveling from Cairo to Moscow via Swiss Air Suppose I am a naturalized citizen of the US, but also a citizen of another country, "XYZ". I then boarded a plane and left the US to another country (maybe not necessarily to XYZ), without first getting a US passport, but using my old, but valid XYZ passport.   In this case what would be the best documents to have for me to gain entry back into the US as a US Citizen without a passport?  Would I need to contact the US Embassy for help in order to be able return?  If the airline does let me aboard to return to the US, even without the US passport, what are the best documents could I show at the border to prove I am a US citizen but that I just don't have a passport, for whatever reason? Why has my database connection failed? Ubuntu n00b here.  I have created several disasters of the first order already.  I am getting ready to try to install opera - it is a very long and arduous process.  I am afraid - no - I am absolutely certain that I will damage something important.  How do I create a system restore point?  Does such a thing exist? I'm from the State of Kuwait and hold an Eritrean passport. I'm planning to visit UK in October but my visa will expire in November.    Is it OK to make a journey in the last month of my visa, provided that I make sure that I book my plane back before expiring date?