Let $f:\ \mathbb{R} \rightarrow \mathbb{R}:\ x \mapsto f(x)$ be a function that is derivable on $\mathbb{R}_0$. Suppose that we know that $\lim_{x \rightarrow 0}f'(x)$ exists and is finite. Can we conclude that $f$ is derivable in $0$?  What I'm trying to prove is that the following limit exists: $$\lim_{x \rightarrow 0}\frac{f(x)-f(0)}{x}$$ I have feeling that we're going to require uniform continuity somewhere. I started out by writing out the limit in full:  $$\lim_{a \rightarrow 0}f'(a) = \lim_{a \rightarrow 0}\lim_{x \rightarrow a}\frac{f(x)-f(a)}{x-a}$$  Now, can I just say the following or are there preconditions? $$\lim_{a \rightarrow 0}\lim_{x \rightarrow a}\frac{f(x)-f(a)}{x-a} = \lim_{x \rightarrow 0}\frac{f(x)-f(0)}{x}$$
I ran across this problem in my Analysis class and can't seem to come up with a good solution. Here's the question: Show that if $\lim_{x\to a}f'(x) = A$ then $f'(a)$ exists and equals $A$. $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$.  I tried applying the Mean Value theorem like this: There exists a $c \in (a, x)$ st $f(x)-f(a)=f'(c)(x-a) \implies \frac{f(x)-f(a)}{x-a}=f'(c) \implies f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}=\lim_{c\to a}f'(c)=A$  However, I think this is wrong. I am pretty sure the Mean Value theorem is needed, but my application of it seems to be incorrect. Would somebody be able to point me in the right direction?
How to restart the networking service? I tried to use  sudo service networking restart   and  sudo /etc/init.d/network restart   but they both crash the window manager and I can no longer use my keyboard for input into X.  when I use the /etc/init.d/ method it complains saying that I should use the service utility  e.g. service networking restart   but it crashes just the same.  Is there a GUI method of restarting networking? What are the “spam” and “rude or abusive” (offensive) flags, and how do they work? Python 3 with QGIS on Mac? Infinite algebraic extension of a finite field I have recently started studying algebraic field extensions and I got to know that algebraic closures $\overline{F}$ of finite fields $F$ are infinite. Therefore, I've asked myself the following question and couldn't come up with an answer.  Suppose $F$ was a finite field, i.e. $|F| &lt; \infty$. Is it possible to construct an algebraic extension $F'$ of $F$ sucht that $|F'| = \infty$ and $F' \subsetneq \overline{F}$? Meaning, that $F'$ is not algebraically closed. List all columns for a specified table Determining the default Method used in optimization and root-finding algorithms In Westeros there are three coins:   Copper Stars (The Seven-Pointed Star is the symbol of the Faith of the Seven.) Silver Stags (The Stag is the sigil of house Baratheon.) Gold Dragons (The Dragon is the sigil of house Targaryen.)   Was the Stag already on the silver coin before Robert became king?  If so, why? Was house Baratheon that important?  If not, what was on the silver coin during the reign of the Targaryen kings?  Why is the sigil of house Targaryen still on the gold coin after Robert took the crown?  Robert hates the Targaryens so much, even murdering innocent children is OK for him.   Keeping this in mind I find it especially strange, that he doesn't change the sigil on most valuable coin. I've got `PDOException: SQLSTATE[HY000] [2002] No such file or directory` when using drush I installed Drush for the first time (so it may be a configuration error) and I'm trying to update to the latest core version of Drupal.  Drush is giving me this error - I'm on an OSX box.  dev5:clientnamedirectory my.name$ drush pm-update projects drupal-7.25 &lt;h1&gt;Additional uncaught exception thrown while handling exception.&lt;/h1&gt;  &lt;h2&gt;Original&lt;/h2&gt;&lt;p&gt;PDOException: SQLSTATE[HY000] [2002] No such file or directory in drupal_get_installed_schema_version() (line 155 of /Users/my.name/Sites/proface/includes/install.inc).&lt;/p&gt;  &lt;h2&gt;Additional&lt;/h2&gt;  &lt;p&gt;PDOException: SQLSTATE[HY000] [2002] No such file or directory in system_list() (line 165 of /Users/my.name/Sites/clientnamedirectory/includes/module.inc).&lt;/p&gt;&lt;hr /&gt;  Drush command terminated abnormally due to an unrecoverable error. Can I bind a (large) block of addresses to an interface? I know that the ip tool lets you bind multiple addresses to an interface (eg, ). Right now, though, I'm trying to build something on top of IPv6, and it would be really useful to have an entire block of addresses (say, a /64) available, so that programs could pick any address from the range and bind to that. Needless to say, attaching every IP from this range to an interface would take a while.  Does Linux support binding a whole block of addresses to an interface? How do I open a terminal? Possible Duplicate:             I just got ubuntu running on my computer and now I am ready to write a program...but I can't figure out how to open a command terminal. I guess I don't know where to find which version I am running either. Any help would be appreciated. Thank you. Why is $ 561 = 3*11*17 $ the smallest Carmichael number? I've been playing Agar for some time now, and lately I've noticed there was a new XP system in Agar. Since then I've been wondering what variables chance the XP gained. I've tried googling it, but with no luck.  I'm looking for how the XP system works in these game modes:   FFA Teams Experimental Why doesn't \makeatletter work inside \newcommand? After hours of banging my brain against my skull, I narrowed down an issue I'm having to the fact that \makeatletter doesn't work inside \newcommand. Commands that I call after \makeatletter are registered as undefined, as if \makeatletter didn't do anything.  Why is this the case?  In the example below, I define two commands. Calling one of them from the document body causes the undefined error, while the other doesn't.  \documentclass[11pt]{book} \usepackage{kvsetkeys}  \newcommand{\commandThatFails}{     \makeatletter \comma@parse{}{} \makeatother }  \makeatletter     \newcommand{\commandThatWorks}{         \comma@parse{}{}     } \makeatother  \begin{document}  Hi there.  \commandThatFails \commandThatWorks  \end{document} Footnote citations in biblatex ignore spacing before the citation command; how can I have the same behavior for all footnotes in a document?  Here's a MWE demonstrating the issue:  \documentclass{article} \usepackage[style=verbose-trad2]{biblatex} \addbibresource{biblatex-examples.bib} \begin{document} Spacing is fine before citations     \autocite{kant:kpv}  but not before footnotes    \footnote{A footnote \autocite{kant:ku}}. \end{document} The curve $x^3− y^3= 1$ is asymptotic to the line $x = y$. Find the point on the curve farthest from the line $x = y$ (NBHM_2006_PhD Screening Test_Analysis)     The curve $x^3− y^3= 1$ is asymptotic to the line $x = y$. Find the   point on the curve farthest from the line $x = y$   how should i solve this problem I am going to Canada, I have a valid U.S. tourist visa (B1/B2). Can I transit via a U.S. airport on the way to Canada with this visa? Logically I know I can since I am allowed to enter the U.S. but I am just making sure. Proposition II.7.5 of Hungerford's Algebra goes     Proposition 7.5. A finite group is nilpotent if and only if it is the direct product of its Sylow subgroups.   Let $G = (\mathbb{Z}_6, +)$. $G$ is abelian, so it is nilpotent and thus by the proposition is the direct product of its Sylow subgroups.  The Sylow subgroups of $G$ are $H = (\mathbb{Z}_3, +)$ and $K = (\mathbb{Z}_2, +)$, so the proposition says $G = H \times K$. In particular, $0 \in H$ and $0 \in K$, so $(0,0) \in H \times K = G$... but that's not true, since $G = \{0,1,2,3,4,5\}$.  Where is/are my misunderstanding(s)? Sorting references with special characters alphabetically I have a problem within the reference list. Namely I have some references where the authors first surname letter is Š (\v{S}) and when listing them in the references they appear amid those authors whose surname starts with V instead after those whose surname starts with S and those whose starts with T (it should list R, S, Š, T... and not R, S, T, U, V with Š in between)  Here is my minimal working example:   \documentclass{article}  \usepackage{filecontents}  \begin{filecontents}{AA.bib}  @article{simovic13, Author = {\v{S}imovi\'{c}, H. and Deskar-\v{S}krbi\'{c}, M.}, Journal = {Zbornik radova Ekonomskog fakulteta u Rijeci}, Number = {1}, Pages = {55--78}, Title = {Dynamic effects of fiscal policy and fiscal multipliers in {Croatia}}, Volume = {31}, Year = {2013}}  @misc{tenhofen07, Author = {Tenhofen, J. and Wolff, G. B.}, Note = {Deutsche Bundesbank Discussion Paper Series 1: Economic Studies No. 14. Frankfurt am Main: Deutsche Bundesbank}, Title = {Does anticipation of government spending matter? Evidence from an expectation augmented {VAR}}, Year = {2007}}  @misc{spilimbergo09, Author = {Spilimbergo, A. and Symansky, S. and Schindler, M.}, Note = {IMF Staff Position Note No. 11. Washington: International Monetary Fund}, Title = {Fiscal multipliers}, Year = {2009}}  @misc{ver11, Author = {Verhelst, S.}, Note = {Egmont Paper, No. 47, Brussels: Egmont Institute}, Title = {The Reform of European Economic Governance: Towards a Sustainable Monetary Union?}, Year = {2011}}  @article{vuckovic10, Author = {Vu\v{c}kovi\'{c}, V.}, Journal = {Privredna kretanja i ekonomska politika}, Number = {125}, Pages = {61--88}, Title = {Political Business Cycles: Do They Exist in {Croatia?}}, Volume = {20}, Year = {2010}}  \end{filecontents}   \begin{document} \nocite{*}   \bibliographystyle{plain} % works % \bibliographystyle{alpha} % works ... % \bibliographystyle{apalike} % works \bibliography{AA} \end{document}   I am thankful for any help!!!  Ana How to convert ISO 8601 date (string) to Date? What's in a name: hyperparameters So in a normal distribution, we have two parameters: mean $\mu$ and variance $\sigma^2$. In the book Pattern Recognition and Machine Learning, there suddenly appears a hyperparameter $\lambda$ in the  regularization terms of the error function.  What are hyperparameters? Why are they named as such? And how are they intuitively different from parameters in general?