If $I = [a,b] $ and suppose $f : I → R $ is increasing in $I$, then $f$ is continuous in $a$ if and only if $f(a)$ = $inf$ $\{f(x):x ∈→ (a,b)\}$ I've solved it this way, okay? If not, please help me correct it. Let $f$ continue en $x=a →$ lim $x_→a^+$ $f(x)=f(a)$ Let's suppose $inf\{f(x)/ x ∈ ]a,b[\}) = L ≠ f(a)→L&gt;f(a)$ lim $_{x→a^+}$ $f(x)=f(a)→$ si $\varepsilon = \frac{f(a)-L}{2}&gt;0, \exists \delta$ $&gt; 0 / a&lt;x&lt;a + \delta →|f(x)-f(a)|&lt;\varepsilon →$ $|f(x)-f(a)|&lt; \frac{f(a)-L}{2}→f(x)-f(a) &lt;\frac{f(a)-L}{2}→ 2f(x)-2f(a)&lt;f(a)-L→$ $f(x)&lt;\frac{3f(a)-L}{2}→ f(x)&lt;\frac{3f(a)-f(a)}{2}→f(x)&lt;f(a)$ contradiction then $ x&gt;a$
Let $f:(a,b)\to \mathbb{R}$ be a strictly increasing function. Does the limit $\lim_{x\to a^+}f(x)$ necessarily exist and is a real number or $-\infty$?  If so, is it true that $\ell=\lim_{x\to a}f(x)\le f(x) \ \ \forall x\in (a,b)$? Please provide proofs.
What is this type of flexible contact strip called? The LCD below is from a cheap digital clock which was (obviously) damaged (broken glass).    The electrical connection between the PCB and the display was realized with this pink strip.    Its flexible...    ...and consists of alternating rows of conductive and insulating material between the pink boarders.    [above image was altered in contrast/brightness to highlight the alternating strips]  What is this kind of connector called? Is it only used for displays?   I am pretty impressed that the connection is reliable just from the very slight pressure between the strip and the PCB. Also, its easily removable from the display. There seems to be some conductive layer on the glass, from where the signal goes into the displays internals. This connection seems to be damaged for the three rightmost signals, where the glass is broken. In Fantastic Beasts: The Crimes of Grindelwald Dumbledore teaches DADA. But from the books we always heard about him as of a Transfiguration teacher (say, in Tom Riddle's days). It is stated in the movie that Dumbledore was prohibited to teach DADA by the Ministry, though the limitations seemed to be canceled at the end of the movie together with the removal of his handcuffs (not completely clear about the teaching though). The strange thing is, however, that Dumbledore was a Transfiguration teacher and even the Head of Transfiguration Department in Hogwarts before Newt's days, as we learn from Pottermore:  Under the guidance of her [McGonagall] inspirational Transfiguration teacher, Albus Dumbledore, she had managed to become an Animagus She [McGonagall] sent an owl to Hogwarts, asking whether she might be considered for a teaching post. The owl returned within hours, offering her a job in the Transfiguration department, under Head of Department, Albus Dumbledore.  That is of course if we accept that Minerva McGonagall in the movie is the same Minerva we know. So it seems that he taught Transfiguration, then for some reason switched to DADA and then to Transfiguration again. While it is possible in general, is there any explanation for the reasons? Confusion between register size, address size, data size I started learning about pointers in C++ and I figured I should educate myself a bit on how memory works and is accessed. I read that when we say a processor is 64-bit, it has a 64-bit register and can access 2^64 address locations. I also read that it can fetch 64/8, so 8 bytes of data at a time, but the data bus is not the same thing as the register right? Do they need to be the same size or does that just happen to be the case usually? And is it the size of the register or the data bus size that determines what we refer to it as, such as 32/64-bit? Also is there a difference between register size and address size? Slave duplicate counter Is it possible to create a counter that is a "slave" to another (master) counter, but also an exact duplicate?  The \@addtoreset{&lt;slave&gt;}{&lt;master&gt;} resets &lt;slave&gt; whenever &lt;master&gt; is incremented (similar functionality is offered by ), but I don't know about anything incrementing one counter (&lt;slave&gt;) automatically whenever another counter (&lt;master&gt;) is incremented.  The idea is to have a macro (say) \dupcntr{&lt;slave&gt;}{&lt;master&gt;} as an analogue to \@addtoreset. The application for this might be where one is interested in renaming a counter used/provided by a package into something more meaningful for the user (say, &lt;slave&gt;), without having to reference &lt;master&gt;. Or, if the original counter contains @ that needs to be escaped every time it's used, it is sometimes more convenient to have a "more user friendly" counter.  I know one way around is to use a macro that returns the value of &lt;master&gt;, but is it possible to perform the same manipulation with a macro than one could with the counter? Manipulation here refers to the printing of the counter (via \the&lt;slave&gt;) and perhaps it's representation (like \arabic, \alph, ...). ref to subsection number only How to solve $x^3=-1$? How to solve $x^3=-1$? I got following:  $x^3=-1$   $x=(-1)^{\frac{1}{3}}$  $x=\frac{(-1)^{\frac{1}{2}}}{(-1)^{\frac{1}{6}}}=\frac{i}{(-1)^{\frac{1}{6}}}$... how to install drivers for wifi adaptor Ralink 148f:7601 on ubuntu? How do I get attention for old, unfixed bug reports and feature requests without official responses here on Meta? root owned program with setuid bit on Ping is a a program owned by root with the user id bit set.  $ ls -l `which ping` -rwsr-xr-x 1 root root 35752 Nov  4  2011 /bin/ping   As I understand it, if a user runs the ping process, then the effective user id will change from the real user id (i.e. the user id of the person who launched the process) to the user id root. However when I try this and look at the output of ps to see if the ping process is running as the root user, I still get the real user id showing.  ps -e -o user,ruser,euser,cmd,args | grep ping sashan   sashan   sashan   ping -i 10 -c 1000 www.goog ping -i 10 -c 1000 www.google.com CASE WHEN with GROUP BY I have a table that looks like this (oversimplified case of the actual case to make my point)  CREATE TABLE sales (    id int,    currency VARCHAR(5),    price int    shop_id int );   Now I need to make a query that will give me something like this table, grouping by shops yet giving sum of sales per currency My problem is when I make currency column non aggregate it fails because my query becomes non standard. When I add currency to group by then each currency creates new row, something I don't want. I have run out of option and would appreciate any help or pointer to the right direction.  I use MySQL but would appreciate if I would get the solution that works with more databases (PostGreSQL being at the top of the list after MySQL)  Query that works with undesired results  SELECT   shop_id,           (CASE WHEN currency= "GBP" THEN SUM(price) ELSE 0 END) AS POUND,           (CASE WHEN currency= "USD" THEN SUM(price) ELSE 0 END) AS USDOLLAR GROUP BY shop_id, currency   The data that should be produced by the query  shop_id     |   POUND   |   USDOLLAR     -----------------------------------------     1       |   400     |   300 -----------------------------------------     2       |   250     |   100 -----------------------------------------     3       |   400     |   100 ----------------------------------------- I would like to replace the search submit button with a Font Awesome character like the Magnifier (Search Icon) character, and put Search the site inside the search box.  How can I achieve something like this? Specifying element type for DXF creation using ogr2ogr I am trying to create a DXF from a Polygon Shapefile using ogr2ogr:  ogr2ogr -f DXF parcels.dxf parcels.shp   The problem is that the polygons are written as hatch elements.    How do I specify that ogr2ogr should create LWPOLYLINE elements (Or any other )?  I have tried this from command line and from QGIS 'save as' option with the same result.  I don't remember this being an issue in the past, but seems like it started with GDAL version 1.10.x. Is it acceptable to change colors on a logo to make it fit a website's theme? I have a few payment brand images that need to go in the footer of a website, the footer is very dark grey, to the point where making the logos grey-scale wouldn't be much of a good idea, especially as we still need the logos to stand out.  For example I took the following logo:   And changed it to fit with our very dominant red theme:    Is this good in terms of the user's perspective? Is this legal or misrepresenting the brand? If so is completely changing a logo to white acceptable for dark backgrounds? Show that all horses are of the same color. Im trying to understand the solution about this, and all the other solutions are.. a bit weird for me and dont answer all my questions!  Reference :   Im using the answer there as reference, but my understanding is a bit different!     Find the error in the following proof that all horses are the same   color. CLAIM: In any set of h horses, all horses are the same color.      PROOF: By induction on h.      Basis: For h = 1. In any set containing just one horse, all horses   clearly are the same color. Induction step: For k > I assume that the   claim is true for h = k and prove that it is true for h = k + 1. Take   any set H of k + 1 horses. We show that all the horses in this set are   the same color. Remove one horse from this set to obtain the set H1   with just k horses. By the induction hypothesis, all the horses in H,   are the same color. Now replace the removed horse and remove a   different one to obtain the set H2 . By the same argument, all the   horses in H2 are the same color. Therefore all the horses in H must be   the same color, and the proof is complete.   Here's what i have come up with :  Suppose a set of 1 horse. $H_a = { horse_{1} }$  Obviously, all the horses in set $H_a$ are the same, since theres just 1 horse.  So using that as the Basis of the induction we can proceed and try to prove its true for $N$ horses.  -- I dont fully understand the part below   As an example : set $H_0$ :   $H_0 = { horse_{1}, horse_{2}, horse_{3}... horse_{n} }$  In my understanding, its because, I can make a subset  formed by 1 horse :  $K_1  = {horse_{1}}$ $K_2  = {horse_{2}}$ ...  I don't get it. I clearly know that this proof is a lie.  And for me, its because you cant claim theres a relation on a set with size $&gt;= 2$ , since :  $H_1 = { horse_{1}, horse_{2}}$  Theres nothing that says $horse_{1}$ equals $horse_{2}$. The basis was for one horse. But that alone is not a proof.  So what should I do ? If $Y|X \sim \mathcal{N}(0,1)$ then is $Y^2|X \sim \chi^2(1)$? Suppose we have random variables $X$ and $Y$ such that $Y|X \sim \mathcal{N}(0,1)$. Can we then say that $Y^2|X \sim \chi^2(1)$? If we can, then what about when $Y|X \sim \mathcal{N}(0,\sigma^2/4)$, can we use the results of  to say that $Y^2|X \sim \frac{\sigma^2}{4}\chi^2(1)$? Essentially I want to know if the results of squaring a normal distribution can be applied to the case where a random variable is conditionally normally distributed? Url decode UTF-8 in Python What is the difference between using a colon and a semi-colon to join two sentences? I've seen a few questions on this site relating to semi-colons, which I believe I correctly understand, but what I'm not as clear about is colons.  For example:     The man ate the apple; it tasted good.   Is this not correctly also written as:     The man ate the apple: it tasted good.   In this case, what is the difference between the two sentences? Hubble's law says that the universe is expanding. How come the Milky Way and the Andromeda galaxies are on a collision course? How will they end up colliding with each other? Finding $\tan t$ if $t=\sum\tan^{-1}(1/2t^2)$ I am solving this problem. If $$\sum\limits_{i=1}^{\infty} \tan^{-1}\biggl(\frac{1}{2i^{2}}\biggr)= t$$ then find the value of $\tan{t}$.   My solution is like the following: I can rewrite: \begin{align*} \tan^{-1}\biggl(\frac{1}{2i^{2}}\biggr) &amp; = \tan^{-1}\biggl[\frac{(2i+1) - (2i-1)}{1+(2i+1)\cdot (2i-1)}\biggr] \\\ &amp;= \tan^{-1}(2i+1) - \tan^{-1}(2i-1) \end{align*}  and when I take the summation the only term which remains is $-\tan^{-1}(1)$, from which I get the value of $\tan{t}$ as $-1$. But the answer appears to be $1$. Can anyone help me on this. State-of-the-art algorithms for the training of neural networks with GRU or LSTM units I recently read a lot about neural networks using GRU or LSTM units. There are many easy to use frameworks like tensorflow that do not even require high knowledge about programming. Unfortunately, I never really found good information on how the training of those networks work. Simple backpropagation might probably not work for gated recurrent networks or is just too inefficient for networks with such a high number of variables to learn.  So my question is:  What are the state-of-the-art algorithms used for the initialization and training of neural networks with GRU or LSTM units? I am not looking for frameworks to use, but for initialization and update equations for the internal parameters.