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Show that function $f$ has a continuous extension to $[a,b]$ iff $f$ is uniformly continuous on $(a,b)$ Let $E \subset F \subset X$ and $f:E\rightarrow Y$. We say that the function $g:F\rightarrow Y$ is an extension of $f$ if $g(x) = f(x)$ for all $x \in E$.   Let $f: (a, b) \rightarrow \mathbb{R}$. Show that $f$ has a continuous extension to $[a, b]$ if and only if $f$ is uniformly continuous on $(a, b)$.   $(X,d)$ is a metric space.  Hint: To prove $\Leftarrow$ start by showing that $f$ maps Cauchy sequences to Cauchy sequences. Then how should you define $g(a)$ and $g(b)$?  The $\Rightarrow$ part of the proof was quite simple, but even with the hint, the other way seems a bit difficult. Doesn't the first part of the "hint" follow from the way we defined $f$? If so, how do I prove this? Help/Hints appreciated! In the Breusch-Godfrey test we use a model  $$ e_t = \varepsilon_t + \beta_1 \varepsilon_{t-1} + \dots+ \beta_p \varepsilon_{t-p}. $$  If we reject the null hypotesis of no serial auotocorrelation of the error, it means that the residuals follow an auto-regressive model of order ($p$).  If I want to avoid this problem, I must add a certain number of lags of the response variable $y$ as regressors in the original model. However, in some cases this method is not useful.   Are there other options to consider? Let $p,q,r$ be integers such that the symmetric sum of fractions $\dfrac{pq}{r}+\dfrac{qr}{p}+\dfrac{pr}{q}$ is an integer. Prove that each of the numbers: $\dfrac{pq}{r},\dfrac{qr}{p},\dfrac{pr}{q}$ is an integer. How to do this? Someone posted but deleted soon. Let $\mathbf{Y}$ be a random vector. Are $k$th moments of $\mathbf{Y}$ considered? I am self-learning on linear model theory right now, and one thing I find surprising is that although $\mathbb{E}[\mathbf{Y}]$ is defined for a random vector $\mathbf{Y} = \begin{bmatrix} y_1 \\ y_2 \\ \vdots \\ y_n\end{bmatrix}$, there is no mention of further moments besides the covariance matrix.  Google searching hasn't turned up much. Are $k$th (raw) moments of $\mathbf{Y}$ considered, or is there a different idea I don't know about?  I am learning from the text  (the TOC starts in p. 17 of the linked file). By "considered," what I mean is is there such a thing as $\mathbb{E}\left[\mathbf{Y}^k\right]$, and if so, how would such a concept be defined? The book I have only covers the first raw moment, and I find it a bit strange that there is no mention of how to define $\mathbb{E}\left[\mathbf{Y}^k\right]$ given my experience in univariate probability, nor do I have the expertise to define it.  Furthermore, if $\mathbb{E}\left[\mathbf{Y}^k\right]$ isn't defined, is there perhaps a related concept that I don't know about that is used instead? How can I improve viewport performance when using Dyntopo? I'm doing a Blendercookie course on how to sculpt a Scifi weapon with dyntopo. The fact is that when some piece reach a certain level of compleiity (about 1.1 million tris) the systems starts to lags too much and is so frustrating.  I've tried the several options:   Disable double sided lighting Enable VBO    I've so also reduced the topology of hidden faces. But that´s how far I've reached.  I'm using a late Imac Intel I5 Nvidia GPU GTX 680mx  any tips out there? Improving how suggested edits are displayed in your activity history Issue 1: I find it a bit irritating that when I click the Improve button and further edit it, even if I mark it as helpful, only Community takes the credit for the approval. I know the topic itself will contain the link to the suggested edit, but my activity history only says that I made an edit, and not approved a suggested edit. The same holds true for marking one as unhelpful. The activity history should say I rejected an edit, and then made an additional edit. I know it's not often, but sometimes I do try to go back to a suggested edit I've rejected, but when I improved and rejected, it's not traceable for me as it doesn't show in my history or in the question's revision history. So if I don't remember what user suggested it, it's lost to me.  This issue also plagues the  which are awarded for approving and rejecting suggested edits, which essentially you have done, you're just not receiving credit for it. I suggest that when I mark an edit as helpful or not helpful, both me and Community be listed as the approvers or rejecters, unless another user is already listed in the appropriate place.  Update: Here is  of the confusion this causes.  Update 2: Also, when improving a post and marking it as "not helpful", it would be nice to be able to  along with it.    Issue 2: Expanding off the idea that users need more notice for suggested edits that get rejected, the activity history page is much too useless when it comes to listing the suggested edits a user has made. Literally all it does is list them. A user still has to click through to each one in order to find out if it was approved or rejected, or even if it's still pending. Take, for example, this screen shot of the suggested edit history:    Is it really necessary to say I suggested a suggested edit? Really? Of course I suggested it, it's a suggested edit. This isn't !  A good way to fix this would be to change the second "suggested" word into one of: pending, approved, or rejected. This would be a quick way to tell which edits have been approved or rejected and which ones are still pending. Users would be able to target out rejected edits and review them for the reasons. Something like this: Linear algebra is a very well-behaved part of mathematics. Soon after you have mastered the basics you got a good feeling for what kind of statements should be true -- even if you are not familiar with all major results and counterexamples.  If one replaces the underlying field by a ring, and therefore looks at modules, things become more tricky. Many pathologies occur that one maybe would not expect coming from linear algebra.  I am looking for a list of such pathologies where modules behave differently than vector spaces. This list should not only be a list of statements but all phenomena should be illustrated by an example.  To start the list I will post an answer below with all pathologies that I know from the top of my head. This should also explain better what kind of list I have in mind. Suppose $f$ is entire and $|f(z)| \leq A + B |z|^{3/2}$. Show that $f$ is a linear polynomial.   My attempt: Since $f$ is entire, we know that it has a Taylor series expansion on the circle $|z| = R$, that is $f(z) = \sum_{n = 0}^\infty a_n z^n$. By using Cauchy's estimate, we can write   $|f^n(0)| \leq \frac{n! M}{R^n}$. Since this holds for any $R&gt;0$, it follows that for $n\geq 2$, $|f^n(0)| \leq 0$. This implies that $f^n(0) = 0$ for $n\geq 2$. Hence $f$ is a polynomial of degree at most $1$.  Edit 1 : On the circle $|z| = R$, $|f(z)|\leq A + B R^{3/2}$.  Thank you very much.  Kindly suggest me if I am wrong? Is there any other apporoach to solve such kind of problems? Prove that between any two different real numbers there is a rational number and an irrational number.  My attempt at a proof: (Throughout this proof it is assumed without any loss of generality that $x &lt; y$)  1.) $x$ is irrational and $y$ is irrational.  In this case the arithmetic mean $\frac{x+y}{2}$ is always an irrational number between $x$ and $y$. Also, if $y$ is irrational then it has a non-terminating decimal expansion and so we can always find a positive integer $n$ such that $x &lt; y - 10^{-n}y &lt; y$. The middle of the three numbers here will have a terminating decimal expansion, and hence it is rational. It also clearly lies between $x$ and $y$. These examples of rational and irrational numbers should also work in the cases:  2.) $x$ is irrational and $y$ is rational.  3.) $x$ is rational and $y$ is irrational.  There is one last case to consider.  4.) $x$ is rational and $y$ is rational.  In this case the arithmetic mean of $x$ and $y$ is rational. An irrational number for example could be the ratio $\frac{ex + \pi y}{e + \pi}$  I'm not sure if this proof is a good method or if it is even valid, so some feedback would be appreciated. When I run the following query in query editor I get this error.  Error     Unknown error parsing query   Query  SELECT Id, OwnerId,Account.OwnerId FROM Contact where OwnerId = Account.OwnerId limit 100   Does anybody know what the problem is? The workings of the Hall effect? I want to ask about the workings of the . Why do the electrons come to rest on the edge of the wire? The magnetic field pushes them up, and the electric field pushes them forward. Shouldn't they reach the top of of the conductor and then stop moving up but continue moving forward?  Why does the magnetic field stop them from continuing to flow? The magnetic force is up so why would it effect their velocity forwards? Installing gcc-7 (and newer packages) in Ubuntu 16.04 Note: The legacy Skype for Linux application (v4.*) is now discontinued and was removed from Canonical’s partner repositories, so this question and its answers are obsolete. Please refer to  of our .  Skype 4.3 is just released with considerable improvements. As a regular Skype user I want to use the latest Skype on my Ubuntu 14.04, but there is no updates from Update Manager. Even in Skype website only Skype 4.3 is available only for Ubuntu 12.04. Is there anyway to use this latest Skype on Ubuntu 14.04 or how much we have to wait for the update? Why can't I tame my ocelot? I'm trying to make a cat house for my zoo on Minecraft. I have some untamed ocelots and I also want a few tamed ocelots. Fish only puts them in breeding mode. How can I know if my post(comment, answer, question) was Flagged? Is it possible to see if my post was flagged. The only way as I see it, if you see your post has got deleted, them may be someone might have flagged it for moderation. But is there other way to see if your post was flagged for moderation.   I am wondering if someone's is repeatedly flagged for posts, maybe he can be assigned a badge(bad one :) )  But right now, I want to know if it is possible to see how many posts of mine were flagged. How can I convert an std::string to a char* or a const char*? Seeking English term for relationship between two members of extended family? Here at the Genealogy and Family History Stack Exchange we see many questions about relationship mapping which is also known as:   relationship calculation kinship terminology etc   If you are just trying to understand how to calculate "cousins and removes" then  provides an excellent guide.  However, if you are seeking an English term for a more complicated relationship between two members of an extended family, rather than asking a question like:          site users are encouraged to first review the answers below.    Note: This Q&amp;A arises from  and is set to  to help make it canonical. How to calculate 95% CI for cumulative incidence rate using R? How can I make calls related to a case display in its Chatter feed? I am using the Service Console in Lightning Experience, and using the Chatter Feed component on the Case layout to display updates to the Case.  If I use the "Log a Call" Quick Action from the Chatter publisher, these calls are displayed in the Chatter feed.  This is the desired behaviour.  However, if I insert a Log a Call record related to the Case any other way - e.g. using the Developer Console, or my CTI system's custom components - the call does not display in the feed.  If I query the record, it looks exactly the same as the record that is created with the Quick Action.  It just seems that the Chatter post is never made in this situation.  I have checked my Feed Tracking settings:   Task has Enable Feed Tracking and Subject checked Case has All Related Objects checked   How can I enable calls created with a custom component (part of a CTI system) to display in the case feed? I have a scenario where i want to calculate sum of column   SAL|CAL|TAG|VAL 12.11"|"1.1"|"2.1"|"1.1 13.11"|"1.1"|"2.2"|"2.2 14.11"|"1.1"|"2.4"|"2.4   My below code not giving output giving output as 0.00  I need output for this column CAL VAL  awk -F'|' '{T+=$2} END { printf "%.2f\n", T }' demo.txt   output :  6.7 5.7