Let $f:[a,b]\rightarrow\mathbb{R}$ be continuous on $[a,b]$ and differentiable on $(a,b)$. Show that if $\lim_{x\rightarrow a}f'(x)=A$, then $f'(a)$ exists and equals $A$. Use the definition of $f'(a)$ and the Mean Value Theorem.  Definition of a derivative:  Let $I\subset\mathbb{R},f:I\rightarrow\mathbb{R}$, and $c\in I$. We say $L$ is the derivative of $f$ at $c$ if $\forall\varepsilon&gt;0,\exists\delta&gt;0$ such that if $x\in I$ and $0&lt;|x-c|&lt;\delta$, then $|\frac{f(x)-f(x)}{x-c}-L|&lt;\varepsilon$. We say $f$ is differentiable at $c$ and denote it as $f'(c)=L$. We can say $f'(c)=\lim_{x\rightarrow c} \frac{f(x)-f(x)}{x-c} = L$  Mean Value Theorem:  Suppose $f$ is continuous on a closed intervale $I:=[a,b]$ and that $f$ has a derivative in the open interval $(a,b)$. Then there exists at least one point $c\in(a,b)$ such that $f(b)-f(a)=f'(c)(b-a)$  Quite honestly I do not see how I could make these two things work together. Also, if we simply replace $f'(x)$ with its limit definition we would have a limit inside of a limit, and I am not sure how to deal with that either.  Any help in pointing me in the right direction would be greatly appreciated.
Let $f$ be a real-valued function continuous on $[a,b]$ and differentiable on $(a,b)$. Suppose that $\lim_{x\rightarrow a}f'(x)$ exists. Then, prove that $f$ is differentiable at $a$ and $f'(a)=\lim_{x\rightarrow a}f'(x)$.    It seems like an easy example, but a little bit tricky. I'm not sure which theorems should be used in here.    ==============================================================  Using @David Mitra's advice and @Pete L. Clark's notes I tried to solve this proof. I want to know my proof is correct or not.  By MVT, for $h&gt;0$ and $c_h \in (a,a+h)$ $$\frac{f(a+h)-f(a)}{h}=f'(c_h)$$ and $\lim_{h \rightarrow 0^+}c_h=a$.    Then $$\lim_{h \rightarrow 0^+}\frac{f(a+h)-f(a)}{h}=\lim_{h \rightarrow 0^+}f'(c_h)=\lim_{h \rightarrow 0^+}f'(a)$$   But that's enough? I think I should show something more, but don't know what it is.
On convergence of Bertrand series $\sum\limits_{n=2}^{\infty} \frac{1}{n^{\alpha}\ln^{\beta}(n)}$ where $\alpha, \beta \in \mathbb{R}$ Study the convergence of $$\sum_{n=2}^{\infty}  \frac{1}{n^{\alpha}\ln^{\beta}(n)}$$ where $\alpha, \beta \in  \mathbb{R}$   I have proved that:   This series diverges when $\alpha \leq 0$. This series converges when $\alpha &gt; 1, \beta &gt; 0$ This series diverges when $0 &lt; \alpha &lt; 1, \beta &gt; 0$ This series converges when $\alpha = 1, \beta &gt; 1$      Question: What happens when $\alpha &gt; 0$ and $ \beta &lt; 0$?   There are other questions on MSE which ask about this series, but this question is distinct because   I would like an argument which does not rely on the integral test for series convergence, and this question considers all real $\alpha$ and $\beta$, while other questions ask only about $\alpha, \beta &gt; 0$, where we can apply Cauchy condensation criterion Are mixed geometry types allowed in shapefiles? Does the shapefile support mixed geometry types (ie. point and polygon all in the same file)? every rank 2 oriented matroid is realizable Why won't my follower follow me after the "A Night to Remember" quest line? At the end of the "A Night to Remember" quest line, my follower came with me when I was teleported to the Misty Grove. Once there, I told my follower to wait.  Now, when I was teleported back to the real world, my follower's not with me: presumably because he's still waiting at the Misty Grove. I've waited for some time and still he hasn't come back.  How can I get my follower back? Determining whether two $k$-chromatic graphs are isomorphic (respecting vertex coloration) Am I being scammed by a sugar daddy? Set value of textarea in jQuery I am attempting to set a value in a textarea field using jquery with the following code:  $("textarea#ExampleMessage").attr("value", result.exampleMessage);   The issue is, once this code executes, it is not altering the text in the textarea?  However when performing an alert($("textarea#ExampleMessage").attr("value")) the newly set value is returned? Compile error: package javax.servlet does not exist Viruses in wine? The Story  I installed wine some time ago, and after trying out some versions of Cinema 4D that were "free", I looked in the comments of the videos, and saw some people complaining about viruses. I got worried, downloaded and installed MalwareBytes, and scanned my "system". Of course, since MalwareBytes was runnning through Wine, it only scanned ~/.wine/C:. It found many viruses/trojans and I opted to delete them. I did so.  Questions:   Assuming there is any malware left, can this malware spread into my OS? Tell me the entire truth on what can happen. I deleted my entire .wine directory and am planning to uninstall wine later. I am the only user out of the three on this system that runs wine programs, but wine is installed for all. Is that enough, or must I do more? Please be specific on what I should do because I want to be 100% sure that there are no viruses.   I am sure that many others have these questions too, so please answer this question to the point.   Additional information:  Wine was installed with sudo-apt get. The exact commands can be found here:   Commands:  sudo add-apt-repository ppa:ubuntu-wine/ppa -y sudo apt-get update sudo apt-get install wine1.6 -y   I did try to install a font that was a ttf font. I tried to convert it to otf with no success. Then, I placed it in the fonts directory directly. I then saw that some people said it was malicious, and I removed it.  This question was flagged as  a duplicate of this question:  It is not the same. This question is more precise, as it specifies two scenarios involving viruses, not risks of viruses. The other question talked about Internet Explorer; this one does not mention the installation of any programs.  Note: wine was installed with sudo privileges, so does that mean it runs with sudo privileges? How can I change the language for dictation on the fly? With English not being my first language, I usually need support for at least two languages when using features like auto-correct and dictation.  Is it possible to toggle the language which is used for dictation via a shortcut?  As of now, I always have to manually navigate through the menus in the System Preferences which if far from convenient. I've tried to see if there are any pre-defined actions in Automator, but found none. I have 1-band GeoTIFF file. The color interpretation for band is gray. I want a color table in order to transform my Col.tiff to Col.jpeg or Col.png.  I tried putting a color table in this tiff via a .vrt:  gdal_translate -of VRT  Col.tiff Col.vrt   then adding in the .vrt:  &lt;ColorInterp&gt;Palette&lt;/ColorInterp&gt; &lt;ColorTable&gt;    &lt;Entry c1="0" c2="0" c3="255" c4="240"/&gt;    &lt;Entry c1="0" c2="0" c3="254" c4="240"/&gt;    &lt;Entry c1="0" c2="0" c3="253" c4="240"/&gt;    &lt;Entry c1="0" c2="0" c3="252" c4="240"/&gt; &lt;/ColorTable&gt;   And translate again :  gdal_translate Col.vrt Col.tiff   But it answered:  Warning 1: Unable to export color table to GeoTIFF file.  Color tables can only be written to 1 band or 2 bands Byte or UInt16 GeoTIFF files.   How can I do it?  For information, there is gdalinfo for the Band:  Band 1 Block=55x18 Type=Float64, ColorInterp=Gray   Description = Something   Metadata:     GRIB_COMMENT=Something[p]     GRIB_ELEMENT=TMP     GRIB_FORECAST_SECONDS=0 sec     GRIB_REF_TIME=1401926400 sec UTC     GRIB_SHORT_NAME=SMTHG     GRIB_UNIT=[p]     GRIB_VALID_TIME=1401926400 sec UTC Can I make a script always execute as root? When doing a clean re-installation of El Capitan (highest supported version) from a bootable USB on an older Mac (MBP, 15-inch, Late 2008), I get this error:  An unexpected error occured while signing in. AMD-Action:authenticate:SP  The error shows up, while logging in with the Apple ID into the App Store in the installer.  I tried with different Apple IDs,  Also tried running these commands in the Recovery Terminal and restarting: csrutil clear csrutil enable Get the last item in an array Pretty self descriptive, without being able to view the source of the noise how can we tell the difference between a sound in front of us and a sound behind us if pitch, volume and distance are all the same.   Also assume that it is not biased to one side more than the other, e.g its not to your left or right but directly in front or behind.   I'm interested in knowing if it is a part of the ear that makes this distinction, or a function of the brain. How do we explain the behavior of \rput, \psGetNodeCenter and saveNodeCoors? Introduction  I am writing a tutorial on a short introduction to PSTricks for newbies. I got obstacles to explain how  a node A (for example), \rput, (!\psGetNodeCenter{A} A.x A.y) and (!N-A.x N-A.y) work by design.  I have read the pst-node documentation as shown below.      And the following is taken from page 35-36:    The manual says:   The transformation matrix will be reset by \psGetNodeCenter. (!N-A.x N-A.y) is not affected by \rput.   Even though I read it many times, I still get confused. The statements are too short.  Problem  Please consider the following figure for the remaining discussion.     Case 1  A node A is defined outside \rput and a dot is translated by \rput.   with (!\psGetNodeCenter{A} A.x A.y)  \pnode(2,2){A} \rput(1,1){\psdots(!\psGetNodeCenter{A} A.x A.y)}  with (!N-A.x N-A.y)  \pnode(2,2){A} \rput(1,1){\psdots(!N-A.x N-A.y)}    The results are different. \rput only affects (!N-A.x N-A.y). Case 2  A node A is defined inside \rput and a dot is translated by \rput.   with (!\psGetNodeCenter{A} A.x A.y)  \rput(1,1){\pnode(2,2){A}\psdots(!\psGetNodeCenter{A} A.x A.y)}  with (!N-A.x N-A.y)  \rput(1,1){\pnode(2,2){A}\psdots(!N-A.x N-A.y)}    The results are identical. \rput affects both. Case 3  A node A is defined inside \rput and a dot is defined outside \rput.   with (!\psGetNodeCenter{A} A.x A.y)   \rput(1,1){\pnode(2,2){A}}  \psdots(!\psGetNodeCenter{A} A.x A.y)  with (!N-A.x N-A.y)   \rput(1,1){\pnode(2,2){A}}  \psdots(!N-A.x N-A.y)    The results are different. \rput only affects (!\psGetNodeCenter{A} A.x A.y).   The MWE is given as follows.  \documentclass[margin=12pt]{standalone} \usepackage{pst-node} \addtopsstyle{gridstyle}{gridlabels=5pt} \psset{saveNodeCoors,linecolor=red} \everypsbox{\color{blue}} \begin{document}  \begin{psmatrix} \begin{pspicture}[showgrid=bottom](3,3)     \rput(1,1){Case 1-A}     \pnode(2,2){A}     \rput(1,1){\psdots(!\psGetNodeCenter{A} A.x A.y)} \end{pspicture} &amp; \begin{pspicture}[showgrid=bottom](3,3)     \rput(1,1){Case 1-B}     \pnode(2,2){A}     \rput(1,1){\psdots(!N-A.x N-A.y)} \end{pspicture} \\ % case 2 \begin{pspicture}[showgrid=bottom](3,3)     \rput(1,1){Case 2-A}     \rput(1,1){\pnode(2,2){A}\psdots(!\psGetNodeCenter{A} A.x A.y)} \end{pspicture} &amp; \begin{pspicture}[showgrid=bottom](3,3)     \rput(1,1){Case 2-B}     \rput(1,1){\pnode(2,2){A}\psdots(!N-A.x N-A.y)} \end{pspicture} \\ % case 3 \begin{pspicture}[showgrid=bottom](3,3)     \rput(1,1){Case 3-A}     \rput(1,1){\pnode(2,2){A}}     \psdots(!\psGetNodeCenter{A} A.x A.y) \end{pspicture} &amp; \begin{pspicture}[showgrid=bottom](3,3)     \rput(1,1){Case 3-B}     \rput(1,1){\pnode(2,2){A}}     \psdots(!N-A.x N-A.y) \end{pspicture} \end{psmatrix} \end{document}   Question  How do I have to explain their behaviors to the newbies as the statements in the documentation seem to be unclear?  Note: As (!\psGetNodeCenter{A} A.x A.y) is identical to (A), (A) will not be discussed here.     If you are a teacher, how do you explain it to your   students? NOTE: Signing up or Logging in via OpenID is no longer officially supported from any of our sites. Currently,  is still on the internet, but we are in the process of completely removing it.    Stack Overflow was an early and strong supporter of . We built our sign up/log in flow around it. We were idealistic and had , but these hopes weren't realized. Over the years people have wondered if  is . We've had to remove support as OpenID providers  or .  The time has come to part ways. The reality is OpenID support has created a ton of complexity in our codebase and the number of users actively using OpenID simply don't justify that cost. Users have spoken with their actions. You prefer Google, Facebook and Stack Exchange (aka email/PW) based account auth.     Of our 9,813,747 accounts around one-tenth of a percent are actively using OpenID (defined as having visited a site in the past 12 months). If you include all the inactive accounts it is still less than 2.9% of all accounts.  So, what is the plan?   DONE - Remove the ability to create a new account with most OpenID providers DONE - Add the ability to  DONE - Convert Launchpad (used on  ) to OAuth DONE - Remove creating new accounts with Steam from  (Steam removed support for OAuth. Only 454 accounts that use Steam have visited Arqade in the past 6 months.) DONE - We will email all users that rely on deprecated OpenID credentials and request that they  DONE - Run an announcement banner for any user who signs in with deprecated OpenID credentials urging them to  DONE - End support for logging in via OpenID  In progress - Remove  (note: only 0.2% of authorized sites in the last year have been non-Stack Exchange sites.)   Obviously, if your account relies on OpenID, then make life easy and go now and .   If you've forgotten what you credentials you use to login, then  to refresh your memory. If you don't see one of the three account types shown in the image below (Google, Facebook or Stack Exchange) then click the "add more logins" button to add one of them to your account.    Remember, SE OpenID accounts are really email/pw accounts handled through complicated OpenID code path. They will be migrated to a simpler email/pw only code path as a part of this change. No extra work required on your part. White screen of death: Fatal error: Allowed memory size of X bytes exhausted I have a issue with my Drupal installation, for example:   when I enable or disable the modules, It redirects me to a blank page, when I create a new content type and save it, redirects me to a blank page, when I add a new view and save it, it redirects me to a blank page, when I try to clear the cache, it redirects me back to a blank page, or in similar cases.   Basically all the confirmation pages redirect me to a white screen. When I refresh it again, it shows me the page. I tried to increase the PHP memory value but it doesn't help.  Are there any other solutions for this?  The error which I'm having:      Fatal error: Allowed memory size of 100663296 bytes exhausted (tried to allocate 8192 bytes) in sites/all/modules/views/plugins/views_plugin_localization_none.inc on line 1 I would like to know how to find a set of coordinates on a small circle some distance from another point.   For example, lets say A = (39.73, -104.98) #Denver  B = (39.83, -106.06) #point approximately 50 nautical miles away from A along the great circle track to C  C = (40.75, -111.88) #Salt Lake City, approximately 323 nautical miles away from A  How do I get the set of orange coordinates when I don't know the angle but I do know the cross track distance?  For concreteness, lets say the outer points near B have a cross track distance of +- 30 nautical miles and the inner points have a cross track of +- 15nm.  And the points near C have a cross track of +- 25 nm.  Also I have an initial bearing from A to C of -1.34 #radians Cannot favorite once up voted I'm not sure whether this is exactly the same bug as  or whether this is a new bug.  If you up vote a question and then attempt to add it to favorites too you get Question is already favorited:    This also happens if you up vote on the web site and then try to favorite it from Android. If you don't up vote or if you favorite it first and then up vote it appears to work fine.