I need help finding the set of continuous functions $f : \Bbb R \to \Bbb R$ such that for all $x \in \Bbb R$, the following integral converges:  $$\int_0^1 \frac {f(x+t) - f(x)} {t^2} \ \mathrm dt$$  I am thinking it could be the set of constant functions but i havent been able to prove it :(   I have also noticed that you can kind of  take any two functions and stick them together (continuously extend one into the other) the resulting function verifies the property in question.  I hope you can provide some insight and thank you .
I need help finding the set of continuous functions $f : \Bbb R \to \Bbb R$ such that for all $x \in \Bbb R$, the following integral converges:  $$\int_0^1 \frac {f(x+t) - f(x)} {t^2} \ \mathrm dt$$  I think it might be the set of constant functions but i havent been able to prove it :(   I was thinking that you can use the stone weiestrass theorem considering the set of continuous functions on a closed  interval(non trivial) ,and a subset which contains the set of continuous functions whose integral above diverges in some point in that interval along with  with the set of constant functions. So in order to solve the problem i need only to prove that if two functions do not meet the condition of the problem then their product does not as well .  I hope you can provide some insight and thank you .
How do I set bold and italic on UILabel of iPhone/iPad? iPhone asks 'Trust This Computer' again (I am on macOS 10.14.6 and iOS 12.4)   I am certain that I have trusted my Mac before. Without the latter trust, I can also access the phone in iTunes.   I have tried reset "Location and Privacy" on my phone. Then   When connecting the iPhone with iTunes running, it asks for the trust and I trust it.  Reboot my Mac and the phone asks for another trust.   Is there some known malware mimics system behavior and steal data from an iPhone?  (I do have installed some pir*ted software recently. That's my fault.) How to use "grep" command to find text including subdirectories I want to find all files which contain a specific string of text. The grep command works, but I don't know how to use it for every directory (I can only do it for my current directory). I tried reading man grep, but it didn't yield any help. What exactly happened with Dr. Crusher's husband and Captain Picard? How do I calculate the run rate in cricket? Always sort Recent Places by date modifed "File Open" dialog is missing sidebar items I'm having the strangest issue since updating to OS X El Capitan. My "Open File" dialogs show the sidebar but all my main (and useful) locations are missing. Take a look at this screenshot. This screenshot shows Keychain Access but this happens in Safari when I'm trying to attach a file to Gmail for example, or uploading an image to Facebook.  What I would like is for the sidebar to show my home directories, such as Desktop, Documents, Downloads, etc.  Help? Astronauts land on planet, met by aliens impersonating their deceased relatives When I was young, I read a chilling story of astronauts coming to an unknown planet and found to their delight  the occupants were all members of their family who were deceased. Each crew member goes off to their respective childhood homes to relive memories.   One astronaut however in the same bedroom as his brother suddenly wonders if aliens may have searched the memories of himself and fellow crew members and used this to gain power over the intruders. His brother asks suspiciously what he's thinking and doesn't believe the answer. The last scene shows astronauts being buried by aliens whose faces melt from human into something terrifying.   Anyone know the name of the story and author? I thought it might be something like Coming Home, but only a guess. How to calculate 95% CI for cumulative incidence rate using R? How to bond/connect wires to neodymium magnets? After reading , I learned it is not advisable to solder a wire to neodymium magnets.  After more reading I found  post describing the use of conductive epoxy to bond electrical wires to neodymium magnets, a very messy process which takes several hours of waiting.  I want to ask this community if they have or suggest better methods to bond/connect a wire to a neodymium magnet to act as a conductive connection? Let $\alpha = \frac{1+\sqrt{-19}}{2}$. Let $A = \mathbb Z[\alpha]$. Let's assume that we know that its invertibles are $\{1,-1\}$. During an exercise we proved that:     Lemma: If $(D,g)$ is a Euclidean domain such that its invertibles are $\{1,-1\}$, and $x$ is an element of minimal degree among the elements that are not invertible, then $D/(x)$ is isomorphic to $\mathbb Z/2\mathbb Z$ or $\mathbb Z/3\mathbb Z$.   Now the exercise asks:     Prove that $A$ is not a Euclidean Domain.   Everything hints to an argument by contradiction: let $(A, d)$ be a ED and $x$ an element of minimal degree among the non invertibles we'd like to show that $A/(x)$ is not isomorphic to $\mathbb Z/2\mathbb Z$ or $\mathbb Z/3\mathbb Z$.  How do we do that? My problem is that, since I don't know what this degree function looks like, I don't know how to choose this $x$!  I know that the elements of $A/(x)$ are of the form $a+(x)$, with $a$ of degree less than $x$ or zero. By minimality of $x$ this means that $a\in \{0, 1, -1\}$. Now I'm lost: how do we derive a contradiction from this? The sum of an uncountable number of positive numbers Claim: If $(x_\alpha)_{\alpha\in A}$ is a collection of real numbers $x_\alpha\in [0,\infty]$ such that $\sum_{\alpha\in A}x_\alpha&lt;\infty$, then $x_\alpha=0$ for all but at most countably many $\alpha\in A$ ($A$ need not be countable).  Proof: Let $\sum_{\alpha\in A}x_\alpha=M&lt;\infty$. Consider $S_n=\{\alpha\in A \mid x_\alpha&gt;1/n\}$.  Then $M\geq\sum_{\alpha\in S_n}x_\alpha&gt;\sum_{\alpha\in S_n}1/n=\frac{N}{n}$, where $N\in\mathbb{N}\cup\{\infty\}$ is the number of elements in $S_n$.  Thus $S_n$ has at most $Mn$ elements.   Hence $\{\alpha\in A \mid x_\alpha&gt;0\}=\bigcup_{n\in\mathbb{N}}S_n$ is countable as the countable union of finite sets. $\square$  First, is my proof correct? Second, are there more concise/elegant proofs? Wilson and Fermat's Theorem Topologist's sine curve is connected I just came across the example of the topologist's sine curve that is connected but not path-connected. The rigorous proof of the non-path-connectedness can be found .   But how can I prove that the curve is connected? To be honest, even intuitively I am not being able to see that the curve is connected. I am thinking if it is proved that the limit point of $\sin(1/x)$ as $x \to 0=0$, then it would be proved. But, why is this true? IMO, this limit doesn't exist. Intuitively also, it seems that the graph would behave crazily and not approach a particular value as a tends to $0.$  EDIT (Brett Frankel): There are a few different working definitions of the topologist's since curve. For the sake of clarity/consistency, I have copied below the definition used in the linked post: $$ y(x) = \begin{cases} \sin\left(\frac{1}{x}\right) &amp; \mbox{if $0\lt x \lt 1$,}\\\ 0 &amp; \mbox{if $x=0$,}\end{cases}$$ I'm reading .     The graph of $f: A \rightarrow B$ is the subset of $A × B$ consisting   of all ordered pairs $(a, b)$ with $f (a) = b$. If $A$ happens to be   $B^n$ for some $n ∈ N$, then we say that $f$ is a function on $B$ and   $n$ is the arity of $f$.   I don't understand the notation for the exponent of a set, I'm aware of what  is, it's the number of arguments or operands the function takes. I'm just not so sure if I can assume that a set with an exponent such as in $B^n$ means that a function has two operands.  I know the question may be too naive and perhapes the answer is the one I suggested and it's right in front of my face, I'm just searching for a confirmation or the negation of it. Thanks. Introduction to statistics for mathematicians What is a good introduction to statistics for a mathematician who is already well-versed in probability?  I have two distinct motivations for asking, which may well lead to different suggestions:   I'd like to better understand the statistics motivation behind many problems considered by probabilists. I'd like to know how to better interpret the results of Monte Carlo simulations which I sometimes do to form mathematical conjectures.   I'm open to the possibility that the best way to go is not to look for something like "Statistics for Probabilists" and just go to a more introductory source. Why was this very low quality flag declined? Question (since closed and deleted): . I know about the  of VLQ flags, and I can't fathom why anyone would think that question wouldn't be a candidate for removal.  Question text, for posterity:     Hey I want a Cyrus beck line algorithm program in c or c++. I have   searched a lot but hadn't got anything over Web..!!      If anyone of you have Program please give it to me and help me out.      Thank you in advance :)   I find  not compelling because it requires me to accept the premise that this question was not "unlikely to be salvageable". Using ArcMap 10.0, I am working on a map that shows many villages. Each village has a name attribute that I want to use for a label, but in some areas there are so many villages that not all can be labeled. I am using the Maplex label engine and have not checked the "Never Remove (allow overlap)" box.  I am quite happy that not all village names are displayed as this would be way too much. And it is ok that Maplex chooses which labels to display, as I have no way to know which village is more important than another one. This also means that I can't base the labeling on a field such as population or something else, even though that would be less random.  However, I would also like the village points that are not labeled to similarly not be displayed. I think it unlikely that someone would want to see the point without knowing the name of the village. If all points are shown, it is also difficult to tell which point a name label goes with in crowded areas. So instead of choosing just labels not to be shown, I also want the computer to chose features not to be shown.  To give you a better idea on what I mean:    Is there a way to make these features/points not draw if their label is not shown? Last two digits of $3^{400}$ No hyper refs from citation to bibliography using BiBTeX, hyperref and harvard