Let $g$ be a function on $\mathbb R$ to $\mathbb R$ which is not identically zero and which satisfies the equation $g(x+y)=g(x)g(y)$ for $x$,$y$ in $\mathbb R$.   $g(0)=1$. If $a=g(1)$,then $a&gt;0$ and $g(r)&gt;a^r$ for all $r$ in $\mathbb Q$.   Show that the function is strictly increasing if $g(1)$ is greater than $1$, constant if $g(1)$ is equal to $1$ or strictly decreasing if $g(1)$ is between zero and one, when $g$ is continuous.
This is part of a homework assignment for a real analysis course taught out of "Baby Rudin."  Just looking for a push in the right direction, not a full-blown solution.  We are to suppose that $f(x)f(y)=f(x+y)$ for all real x and y, and that f is continuous and not zero.  The first part of this question let me assume differentiability as well, and I was able to compose it with the natural log and take the derivative to prove that $f(x)=e^{cx}$ where c is a real constant.  I'm having a little more trouble only assuming continuity; I'm currently trying to prove that f is differentiable at zero, and hence all real numbers.  Is this an approach worth taking?
How to run ubuntu/xubuntu in a complete non-gui mode A story about the world split into three realms This is a series I read partially something like 15 years ago (I think). It is set in a universe with magic of some sort - I think the magic was derived from geological processes (?). One of the things I remember - which was the reason I abandoned the series - was that it grew ever more gory; when I stopped reading, one of the main protagonists was flayed alive.   I only remember fragments - there were hints that the universe had been magically split into three or four parts, three relatively normal parts and The Void. I remember there was a powerful person/ruler called Tensor, and a kind of monstrous race with powerful magic, called 'Lorsk', which originated in the void and had evolved from humans that had been trapped there.  I'm a bit ambivalent about reading the series again - it seemed to be a never-ending death spiral of soap operatic proportions, but I'd like to see if my impression from then was actually right. Display header and footer in all PDF pages using VF My vf page     &lt;apex:page renderAs="pdf" Controller="testpdf1"&gt;     &lt;head&gt;         &lt;style type="text/css" media="print"&gt;             @page              {                 size: 8.5in 11in;/* width height */             }              @page              {                 @top-center                  {                     content: element(header);                 }                  @bottom-left                  {                     content: element(footer);                 }             }              div.header              {                 padding: 10px;                 position: running(header);             }              div.footer              {                 display: block;                 padding: 5px;                 position: running(footer);             }              .pagenumber:before              {                 content: counter(page);             }              .pagecount:before              {                 content: counter(pages);             }         &lt;/style&gt;     &lt;/head&gt;      &lt;div class="header"&gt;             &lt;div align="right"&gt;Header&lt;/div&gt;     &lt;/div&gt;      &lt;div class="footer"&gt;         &lt;div&gt;Page &lt;span class="pagenumber"/&gt; of &lt;span class="pagecount"/&gt;&lt;/div&gt;     &lt;/div&gt;      &lt;div class="content"&gt;         &lt;p&gt;             &lt;apex:repeat value="{!opp}" var="item"&gt;                 &lt;tr&gt;                     &lt;td class="tableContent"&gt;{!item.Name}&lt;/td&gt;                 &lt;/tr&gt;             &lt;/apex:repeat&gt;         &lt;/p&gt;     &lt;/div&gt; &lt;/apex:page&gt;   Controller  public class testpdf1 {     public list&lt;Account&gt; opp{get;set;}      public testpdf1()     {         opp=[SELECT Name  from Account limit 50];     } }   I am getting all records in pdf format but able to get header in first page only and i am not getting header from second pages how can i display header to all pages. How do I update my graphics drivers? I am using a laptop and ubuntu 11.04. I would like the easiest way possible because I am new to linux.  here is my display information:  *-display              description: VGA compatible controller              product: Core Processor Integrated Graphics Controller              vendor: Intel Corporation              physical id: 2              bus info: pci@0000:00:02.0              version: 18              width: 64 bits              clock: 33MHz              capabilities: msi pm vga_controller bus_master cap_list rom              configuration: driver=i915 latency=0              resources: irq:41 memory:d0000000-d03fffff memory:c0000000-cfffffff ioport:e080(size=8)         *-communication UNCLAIMED              description: Communication controller How can I make a macOS Mojave bootable USB installer in Windows 10? I have downloaded and saved macOS Mojave installation file from the Mac App Store into my external hard drive. Unfortunately, macOS has crashed on my Mac and now I need to install it again. I have used a software named TransMac to make a installation USB in windows following the steps in the link below:    But the provided USB drive after all these steps doesn't show up in my MacBook pro. Considering that I have saved Install macOS Mojave.app on my external hard drive, and I don't have access to any other Mac computers, How can I fix this issue and build a USB for installing Mojave? Formation of ammonia from air and moisture On extended bicycle trips I sometimes slept under a plastic sheet, which helped with mosquitoes on hot nights. But there was a noticeable build-up of ammonia which made this almost impossible for more than an hour or so at a time.   Air is mostly nitrogen $(N_2)$ and there is always water (containing hydrogen) on a humid day. But I wonder about the mechanism for the formation of ammonia under these circumstances? There are (according to Wiki) trace amounts of methane and $H_2$ in the air. But methane and hydrogen (g) are I think pretty stable.   I wonder if the presence of exhaled air (containing $CO_2)$ could be a factor?  Thanks for any ideas. Undefined reference to static class member PCA on correlation or covariance? What are the main differences between performing principal component analysis (PCA) on the correlation matrix and on the covariance matrix? Do they give the same results? Illustrating op amp feedback without control theory We are teaching op amps in a high school class, before calculus.  So we can't use control theory to teach how op amps respond.  All the same, I would like to have an intuitive explanation for how a feedback circuit works.  Take the negative feedback, for instance.  Is there a clear way to show how an initial difference of delta between V+ and V- leads to a very large (G(V+ - V-)) difference in the output, which then leads to ...  I'd like to be able to take that argument and show how the voltage of the output converges in addition to the standard virtual short argument.  Can anyone explain it clearly? Can helium disappear from Earth? Travelling as an unmarried Western couple in Indonesia I understand that the vibration modes of an elementary string determines the identity of its particle. When I first heard this, I visualized a stiff circular steel band-like entity vibrating with 2, 3, 4, 5,... nodes or antinodes, the number determining the particle. This was further supported when I asked a well-known physicist how one gets from equations to physical strings, and he said he recognized the harmonic nature of the solutions.  Bu then I noticed that physicists seem to accept the image of a soft spaghetti-like string flopping about with no well-defined vibrational modes. I asked another physicist about this, and his answer was simply that a string has multiple modes, not necessarily in a harmonic progression.  So my questions are 1) How do you visualize the vibrations of a string, and 2) how are they related to particle identity? Assume $x$ doesn't occur free in $\alpha$, show that: $$\vdash (\forall x \beta \to \alpha) \leftrightarrow \exists x (\beta \to \alpha)$$   This is an exercise on page 130, A Mathematical Introduction to Logic, Herbert B. Enderton（2ed).   Here's my attempt: For the '$\to$' direction, since $x$ doesn't occur free in $\alpha$, we have $α→∀x α$ and $\forall x \alpha \to \alpha$ as logical axiom(In the second formula, we have$\alpha_x^x = \alpha$). Thus it suffice to show $\vdash (\forall x \beta \to \forall x \alpha) \rightarrow \exists x (\beta \to \alpha)$. By logical axiom $∀ x(α→β)→( ∀ x α→∀x β)$, it would suffice to show $\vdash \forall x (\beta \to   \alpha) \rightarrow (\exists x (\beta \to \alpha))$. It has been shown in a previous exercise that there's a deduction $∀ x ϕ→∃x ϕ$, so we are done for the "$\to$" part.  The problem for me is how to show the "$\leftarrow$" direction.  Added: Here's logical axioms that we can employ:         Tautologies(in the sense of propositional logic);   $∀ x α →α^x_ t $, where $t$ is substitutable for $x$ in $α$ ($α^x_ t$ is the formula derived from $\alpha$ by replacing $x$ by a term $t$);   $∀ x(α→β)→( ∀ x α→∀x β)$;   $α→∀x α$, where $x$ does not occur free in $α$.      Besides, generalization theorem and deduction principle can be utilized:     If $\Gamma \vdash ϕ$ and $x$ do not occur free in any   formula in$\Gamma$, then$\Gamma \vdash ∀ x ϕ$.      If $\Gamma ; γ \vdash ϕ,$ then $\Gamma \vdash  γ \to ϕ$. GUI for watching logs (tail and grep) Could you recommend a GUI application with powerful log watching capabilities?  Generally it would work as tail -f in GUI, but on top of that following features would be very useful:   filtering out some lines based on (regular) expressions  coloring some lines based on (regular) expressions interactive search saveable configuration easily applicable to different files notifications based on (regular) expressions   A similar tool on Windows is  and its paid version - What is causing different spacing between a header and content on one page only? I have found that for only one page (so far) the spacing between the header and content is different from the spacing on all the other pages. You can see this behavior in the images below (from pages 2 and 3 of the document):     My .tex file contents are shown below.  \documentclass[a4paper]{report} \usepackage{fullpage} \usepackage{lipsum} \usepackage{fancyhdr} \pagestyle{fancy} \setlength{\headsep}{0.5cm}  \begin{document} \renewcommand{\headrulewidth}{0.4pt} \renewcommand{\chaptermark}[1]{\markboth{\thechapter. #1}{}} \fancyhead[L]{\bfseries \leftmark} \fancyhead[R]{\bfseries \rightmark}  \chapter{Version Control} \lipsum \lipsum \lipsum \lipsum  \end{document} Why do the big nuts always remain at top? The Brazil-nut Effect Sum of Cosine/Exponential delaying moldy bread I always keep bread around, home-made or otherwise.  Is there a way to keep it from molding, or at least delay the process?  I've tried the fridge with some success (extra day or two), but I don't like my bread that cold to eat.  I've tried zip-top bags with no success (seems to mold faster).  Bread-box didn't seem to make any difference in time.  Thanks There is no set of all sets.. So there are a few posts about this already, but they skip over the problem I have.  The proof of Cantor's theorem elegantly shows that if we consider $f:A\rightarrow \mathcal{P}(A)$, the set $B=\{x\in A : x\not\in f(x) \}$ by its definition cannot lie in the image of $f$, precluding it from being a bijection - hence power sets have strictly greater cardinality.  But now, I thought one (obvious) way to characterise the "set of all sets" would be $S=\mathcal{P}(S)$, as indeed, it contains all possible sets (perhaps here is the problem, but I don't immediately see how using this as a definition for the set of all sets is bad). Now consider simply taking the identity function $i:S\rightarrow S$. Then $B=\{x\in S : x\not\in x \}$ - the set of all sets which do not contain themselves. But this doesn't exist. It's exactly the set of Russell's paradox. If this isn't a set, and so is not in $S$, then the proof no longer offers an obstruction to $i$ being surjective.    Each other post I've seen on the matter uses Cantor's theorem to show such a set doesn't exist. My query is to find that something else that must be happening, since I think the proof has issues for this particular set. I'm expecting other issues to arise with the definition $S=\mathcal{P}(S)$... What's the max height of blocks for a redstone contraption to still work even I'm in the surface below?  Like if I build a pillar to the max height of blocks and build a redstone contraption there, is it still going to work even I'm on the surface below?