It is given a continuous function $f$ on $[0,1]$. Assuming that $$\int_{0}^{1} x^n f(x) \ dx = 0$$ for every $n\in\mathbb{N}$, how to prove that $f(x) = 0$ on $[0,1]$?
As the title says, I'm wondering if there is a continuous function such that $f$ is nonzero on $[0, 1]$, and for which $\int_0^1 f(x)x^n dx = 0$ for all $n \geq 1$. I am trying to solve a problem proving that if (on $C([0, 1])$) $\int_0^1 f(x)x^n dx = 0$ for all $n \geq 0$, then $f$ must be identically zero. I presume then we do require the $n=0$ case to hold too, otherwise it wouldn't be part of the statement. Is there ay function which is not identically zero which satisfies $\int_0^1 f(x)x^n dx = 0$ for all $n \geq 1$?  The statement I am attempting to prove is homework, but this is just idle curiosity (though I will tag it as homework anyway since it is related). Thank you!
Difference between modelAttribute and commandName attributes in form tag in spring? In the recent post , Werner helped me to redefine a proof environment which can be hidden by an optional boolean argument (showProof). Without an argument, somehow the line/paragraph breaking is not correct. How can one fix this?  \documentclass{scrartcl}  \usepackage{ifthen} \usepackage{environ} \usepackage{amsthm}  \newboolean{showProof}% for showing proofs \setboolean{showProof}{true}% setting the switch \newif\ifstarttheorem  % proof environment (with hide feature) \makeatletter \providecommand{\env@proof@save@env}{} \providecommand{\env@proof@process}{} \RenewEnviron{proof}[2][showProof]{   \gdef\@tmp{#1}%   \ifthenelse{\boolean{#1}}{\par\pushQED{\qed}%     \normalfont\topsep2\p@\@plus2\p@\relax     \trivlist\item[\hskip\labelsep\sffamily\bfseries Proof~#2]\gdef\mycurrenvir{proof}\global\starttheoremtrue\mbox{}\hfill\\*\ignorespaces     \BODY}{\ignorespaces} }[\ifthenelse{\boolean{\@tmp}}{%   \gdef\mycurrenvir{\relax}   \popQED\endtrivlist\@endpefalse }{\ignorespacesafterend}] \makeatother  \begin{document} Just some text \begin{proof}   Just some text in a standard proof. As we can see, there is a problem. \end{proof} \begin{proof}[false]   Proof with optional argument set to \texttt{false}. This proof correctly does   not appear. \end{proof} \begin{proof}[showProof]{(Heading)}   Proof with optional argument (\texttt{true}) and second argument ``(Heading)'' \end{proof} \end{document} How to plot periodic function's graphic? I have built a peak detector circuit using op-amps/comparators to amplify the signal from an electret microphone, hold the peak signal and create a sharp logic edge when a peak is detected to be sent to a microcontroller.  I have built the circuit as follows setting up OA1 as a non inverting amplifier with a gain of 1000, but the voltage out from the OA1 remains fixed at about 4.5V no matter how much noise I make into the microphone. The comparator part of the circuit is outputting approximately 2V low and 5V high depending on how I set the potentiometer. All the components appear to be working.         &ndash; Schematic created using After reinstalling TeXLive distribution (install-tl-20160318, Ubuntu 14.04), a issue occurs with the updated version of the titlesec package: the section numbering is lost.  What could it be missing? Is there a modification I have not seen?  Below is a MWE comparison of the two versions. The prevtitlesec package is a reinstall of the previous version (2011/12/15).  \documentclass{book} \usepackage[margin=2cm,a4paper]{geometry} %\usepackage{prevtitlesec}% 2011/12/15 version \usepackage{titlesec}% 2016/03/15 version \usepackage{lipsum}  \begin{document}  \chapter{Title of the first chapter} \lipsum[1] \section{A test section} \lipsum[4] \subsection{A test subsection} \lipsum[4] \subsection{A test subsection} \lipsum[4]  \end{document}   titlesec package 2011/12/15:    titlesec package 2016/03/15: What is the best way to "carve" a terrain created from a heightmap? I have a 3d landscape created from a heightmap. I'd like to "carve" some holes in that terrain. That will allow me to create bridges, caverns and tunnels inside it.    That operation will be done in the game editor so it doesn't need to be realtime. In the end, rendering is done using traditional polygons.  What would be the best/easiest way to do that ? I already think about several solutions :  Solution 1  1) Create voxels from the heightmap (very easy). In other words, fill a 3D array like this : voxels[32][32][32] from the heightmap values.  2) Carve holes in the voxels as i want (easy too).  3) Convert voxels to polygons using some iso-surface extraction technique (like marching cubes).  4) Reduce (decimate) polygons created in 3).  This technique seems to be the most promising for giving good results (untested). However the problem with marching cubes is that they tends to produce lots of polygons thus reducing them is mandatory. Implementing 4) also seems not trivial, i have read several papers on the web and it seems pretty complex. I was also unable to find an example, code snippet or something to start writing an algorithm for triangle mesh decimation. Maybe there is a special decimation algorithm (simpler) for meshes created from marching cubes ?  Solution 2  1) Create some triangle mesh from the heighmap (easy).  2) Apply severals 3D boolean operation (eg: subtraction with a sphere) to carve the mesh.  3) apply some procedure to reduce polygons (optional).  Operation 2) seems to be very complex and to be honest i have no idea how to do that. Also applying many boolean operation seems to be slow and will maybe degrade the triangle mesh  every time a boolean operation is applied. Can I cast a spell while another spell or ability is resolving? My opponent has a  with scads of loyalty. I've got a  and a  on the battlefield.  He activates Sorin's "ultimate" ability, targeting Gideon.     -6: Destroy up to three target creatures and/or other planeswalkers. Return each card put into a graveyard this way to the battlefield under your control.   Is there any way I can use the Spellbomb to prevent my opponent from stealing Gideon after he is sent to my graveyard? equivalent norms in Banach spaces of infinite dimension What is the difference between "accumulative" and "cumulative"? I'm having a hard time understanding the real distinction between  and .  accumulative     adjective         tending to accumulate or arising from accumulation; cumulative.   tending to accumulate wealth; acquisitive.      cumulative     adjective         increasing or growing by accumulation or successive additions: the cumulative effect of one rejection after another.   formed by or resulting from accumulation or the addition of successive parts or elements.   of or pertaining to interest or dividends that, if not paid when due, become a prior claim for payment in the future: cumulative preferred stocks.      These definitions (and those from other dictionaries as well) say different things but I can't understand how they aren't really meaning the same thing.  Yet, when thinking about it, it seems there are some cases where it is distinctly more correct to use one or the other.  What's the difference, practically? How to show that Klein four-group is a normal subgroup of the alternating group $A_4$ I want to show that the Klein four-group is a normal subgroup of the alternating group $A_4$.   I am using the information in , that shows explicitly $A_4$, and Klein four-group as a subgroup.  I know that there is the direct way, by definition, but is there a way that does not require actually multiplying so many permutations ? Whenever I try to do a fluid simulation, the fluid domain remains visible.  I have tried:ctrl+a-scale, turn on/off visibility for render/viewport  Here's a video demonstrating my problem:   Here's the blend file in case I have messed up some settings:   Thank you for any help! $\phi \times \theta$ is faithful iff $(|Z(H)| ,|Z(K)|)=1$ for faithful characters $\phi \in Irr(H)$ and $\theta \in Irr(K)$ . For $a,b,c&gt;0$, Find $$ \lim_{n \to \infty}\left(\frac{a^{1/n}+b^{1/n}+c^{1/n}}{3}\right)^n$$  how can I find the limit of sequence above?  Provide me a hint or full solution. thanks ^^ My colleague and I were discussing about certain types of customers in the IT industry. You have to work extremely hard to convince them so that they think about opening their wallets .  However, they should not be mistaken to be stingy (carries a negative connotation). Cost-conscious is a closely related term but is a bit wordy. Another word that I considered is shrewd but I don't know if it fits my context well.  Usage Sentence:     Nowadays, IT customers are becoming increasingly ________. They demand more for less and probably think a zillion times before reaching for their pockets. If they are not convinced 100%, they simply won't open their wallets.    Is there is a noun or an adjective to describe such people? I am also open to using idioms, if there is one that conveys the intentions well.  N.B:  I am looking for words with neutral to positive connotations. When should "farther" and "further" be used? Will an object always fall at an infinite speed in a black hole? Continuous versus differentiable A function is "differentiable" if it has a derivative. A function is "continuous" if it has no sudden jumps in it.  Until today, I thought these were merely two equivalent definitions of the same concept. But I've read some stuff today which seems to be claiming that this is not the case.  The obvious next question is "why?" Apparently somebody has already asked:    Several answers were given, but I don't understand any of them. In particular, Wikipedia and one of the replies above both claim that $|x|$ has no derivative. Can anyone explain this extremely unexpected result?  Edit: Apparently some people dislike the fact that this is non-obvious to me. To be clear: I am not saying that the result is untrue. (I'm sure many great mathematicians have analysed the question very carefuly and are quite sure of the answer.) I am saying that it is extremely perplexing. (As a general rule, mathematics has a habit of doing that. Which is one of the reasons why we demand proof of everything.)  In particular, can anyone explain precisely why the derivative of $|x|$ at zero is not simply zero? After all, the function is neither increasing nor decreasing, which ought to mean the derivative is zero. Alternatively, the expression  $$\frac{|x + a| - |x - a|}{a}$$  becomes closer and closer to zero as $a$ becomes closer to zero when $x=0$. (In fact, it is exactly zero for all $a$!) Is that not how derivatives work?  Several answers have suggested that the derivative is not defined here "because there would be a jump in the derivative at that point". This seems to assert that a continuous function must never have a discontinuous derivative; I'm not convinced that this is the case. Can anyone confirm or refuse this argument? Every finitely generated algebra over a field is a Jacobson ring Knowing that the polynomial ring in $n$ variables over a field $k$ is a Jacobson ring, how can we prove from it that every finitely generated $k$-algebra is a Jacobson ring?   EDIT: We define a ring to be Jacobson if every prime ideal is an intersection of maximal ideals. After my suspension period, my reputation was not returned On Stack Overflow, I have a suspension period of 7 days which ends today. But my actual reputation has not been returned to me as one of the moderators said it would. I have a message like this by @Andrew Barber:     Once your suspension period is over, you will be returned to your   correct reputation level, minus the votes from the accounts which have   been deleted. At that time, you are more than welcome to return, and   continue contributing your knowledge, and gaining legitimate   reputation points. Just please keep our rules in mind when you do.      Regards      Andrew Barber, Stack Overflow moderator   Can any moderator here help me or tell me what the rules are of getting back my reputation?  How can I get the attention of a moderator so that they can resolve my problem? I want to write a query like this:  SELECT o.OrderId, MAX(o.NegotiatedPrice, o.SuggestedPrice) FROM Order o   But this isn't how the MAX function works, right?  It is an aggregate function so it expects a single parameter and then returns the MAX of all rows.     Does anyone know how to do it my way?