How can I prove that if $f$ is continuous on $[0, 1]$ such that for every $x \in [0, 1]$ we have that $\int_{0}^{x} f(t) \,dt = \int_{x}^{1} f(t) \,dt$, then $f(x)=0$ for every $x$.
If $f:[0,1]\to\mathbb R$ is continuous and $\int_0^x f(t)dt = ∫_x^1f(t)dt$ then prove that $f(x)=0$.  I suppose that $f(x)$ isn't zero, set $g(x)= \int_y^xf(t)dt$ with $y\in[0,1]$ and use Bolzano but I don't know if it's right.
Character Limit in HTML given a polynomial $p(x,y)$ from $\mathbb Z[x,y]$. I want to substitute $x=1$ leaving $y$ as it is. The command Value(p,[1,y]) does not work. Can you give me a hint?  After that it gets a little more complicated. Let  $$p(x,y)=x^6+y^6$$ After substituting $x=1$ we have $p(1,y)=1+y^6$. Given a natural number $j$. I want to find all expressions $y^j$ in $p$ and substitute these $y^j$ by $1$ and the others by $0$. In this case (and for $j=6$) we would get $1+1=2$. If $$p(x,y)=x^4y^2+x^2y^6$$ we should get $0+1=1$. Is there any way to do this?  Thanks for reading! edit: Well, it turns out I used the function Value wrong. The right code for my problem would be Value(p,[x,y],[1,y]). So my first question is solved. Create UEFI bootable USB stick to install Windows AND Ubuntu from it I know how to make a UEFI bootable stick to install Ubuntu (explained ) and I know how to make one to install Windows 7 (, sorry). I guess for Win 8 the procedure would be similar, but I haven't looked into it.  I also know how to make a legacy bootable stick from which I can install Windows and Ubuntu using YUMI. I have failed so far though at creating a UEFI bootable stick that can do this. When I do a UEFI boot from my YUMI multiboot stick it gives me no option to choose between Ubuntu and Windows (hardly surprising I guess) and instead simply UEFI boots Windows (which I find kind of surprising to be honest but maybe this is due to the fact that I added the Windows image after the Ubuntu image and this simply replaced Ubuntu's EFI boot files).  So how would I proceed to create such a bootable stick? When I turn 13 years old, will some of my Roblox account restrictions unlock automatically? `/usr/bin/cd` is not working in mac terminal Google isn't surfacing Events rich snippets in SERP when JSON-LD Structured Data is valid Google confirms our site has thousands of valid MusicEvent structured data objects presented using JSON-LD, as indicated in the Structured Data section of Webmaster Tools. However, rich snippets for the events do not show up in any search result pages. The Rich Cards tab in Webmaster Tools says:     We did not find any structured data for rich cards on your website.   The structured data is valid with no errors in the Structured Data Testing Tool. We do generate some warnings occasionally for missing endDate and price fields. Comparing our code to a survey of other sites whose rich snippets do appear, we are not doing anything unique in terms of our markup.  The data has been there for months, in fact over the months we've tried also using Microdata format. The Structured Data section indicates that many pages were crawled this week.  Besides SDTT validation, what else can I check to troubleshoot this issue? If $A$ is a ring and $S=\{1,f,f^2,f^3,...\}$ a multiplicative set of $A$, prove that $\mathrm{Spec}(A_f)=\mathfrak{V}((f))^c$.   Notation: $A_f=S^{-1}A$ and $\mathfrak{V}((f))=\{P \in \mathrm{Spec}(A): P \supset (f)\}$  My attempt:   On one hand, if $P \in \mathrm{Spec}(A)$ and $P \cap S$ is empty then we identify $\mathrm{Spec}(A_f)=\mathrm{Spec}(S^{-1}A)=\{P \in \mathrm{Spec}(A): P\cap S= \emptyset\}$.  On the other hand, $\mathfrak{V}((f))^c=\{P \in \mathrm{Spec}(A): P \nsupseteq (f)\}$.   So the exercise is equivalent to prove: $$\{P \in \mathrm{Spec}(A): P\cap S =\emptyset\}=\{P \in \mathrm{Spec}(A): P \nsupseteq (f)\},$$ but I can't continue so I'd appreciate if somebody could help me.   Thanks in advance. Set myself as owner of /etc with chown command now getting all kinds of errors Electromagnetic waves in vacuum If there is no source then also there is electromagnetic waves described by Maxwell equation. how if there is no source then existence of EM waves. What gives energy to this EM  waves. Is it vacuum fluctuation or something else? Let $f(x)$ a irreducible polynomial of degree $n$ over a field $F$. Let $g(x)$ be a polynomial in $F[x]$. Prove that every irreducible factor of the composition $f(g(x))$ has a degree which is divisible by $n$.  I don't know even how to begin. I really need help. Thanks How do I survive in the Nether? Whenever I travel to the Nether in Minecraft, I almost instantly get blasted to smithereens and chopped up into little pieces.  How do I survive in the Nether, at least long enough to build some sort of shelter and get situated? What do I use on linux to make a python program executable It seems that a List object cannot be stored in a List variable in C#, and can't even be explicitly cast that way.  List&lt;string&gt; sl = new List&lt;string&gt;(); List&lt;object&gt; ol; ol = sl;   results in Cannot implicitly convert type System.Collections.Generic.List&lt;string&gt; to System.Collections.Generic.List&lt;object&gt;  And then...  List&lt;string&gt; sl = new List&lt;string&gt;(); List&lt;object&gt; ol; ol = (List&lt;object&gt;)sl;   results in Cannot convert type System.Collections.Generic.List&lt;string&gt; to System.Collections.Generic.List&lt;object&gt;  Of course, you can do it by pulling everything out of the string list and putting it back in one at a time, but it is a rather convoluted solution. A stranger told me that he will give me an allowance of $ weekly. I know the saying "If it sounds too good to be true, then it is".   He told me that he would  email me his check and I would be able to deposit it via mobile banking. The only information he needs is my email, name on bank account, and bank name.   Is this too much information? Do you think this is a scam? I would definitely know if it was a scam if he asked for my log in but that is not what he wants.  Thank you for your time and answers! How can I build XML in C#? How can I generate valid XML in C#? Show that $d_2$ defined by $d_2(x,y)=\frac{|x-y|}{1+|x-y|}$ is a metric I am pretty sure that this is such a stupid, stupid question, but how do you prove that $d_2(x,y)={|x-y|\over {1+|x-y|}}$ satisfying the third condition to be a metric, which is the triangle inequality.   I know that I should really be able to do this, but I just can't, I keep seeing a contradiction. Please help me. Why does main method in Java always need arguments? What is the recommended CIDR when creating VPC on AWS? I have been creating AWS VPCs and I am wondering if there is a recommended CIDR value when creating VPCs. What are the factors that I must consider when choosing a CIDR and does the CIDR value affect the performance of the network? G-test vs Pearson's chi-squared test Is there an operation similar to inset that creates faces in the opposite direction? That is, instead of creating faces inwards, they create faces outwards? A normal inset operation:  What I would like: