let $f: J \to \mathbb{R}$ be differential function . let $x_n &lt; c &lt;y_n$ be such that $(y_n-x_n) \to0$  show that     $$\lim _ {n \to \infty}  \frac{f(y_n)-f(x_n)}{y_n-x_n}=f'(c)$$   i am trying to apply Mean value theorem on $f$ on the interval $[y_n,x_n]$.this gives   $$\frac{f(y_n)-f(x_n)}{y_n-x_n}=f'(\theta_n)$$  Now as $n$ goes to infinity $\theta_n \to c$. but $f'$ is not given to be continuous.if that was given then  we had done.  But how to approach this problem  since $f'$ is not given to be continous . Any Hint.
Let $f$ be differentiable at $x=a$.   Prove that if $x_n \to a^+$ and $y_n \to a^-$ then:  $$\lim_{n\to \infty} \frac{f(x_n)-f(y_n)}{x_n-y_n}=f'(a).$$  Every option that I think about seems to my very trivial, so I believe that I am doing something wrong.  Both numerator and denominator approach zero as $n\to\infty$ as the case of the formal definition of derivative, but it isn't guaranteed that the limits are equal (“$\frac{0}{0}$”).   Any direction?
I can't distinguish the difference in meaning between these  two sentences.   It really doesn't matter.  It doesn't  really matter.    It seems that there is a nuanced difference, but I cannot see what this is. Changing default width of blocks in beamer I use a custom Beamer theme for my presentations. In my presentations I'd like to use blocks (\begin{block}…\end{block}). These blocks are rendered slightly too wide to fit into my theme. Is there a way how I can set the default width of blocks, e.g. to .9\textwidth? Prove that the function : $f(x) = x$ when $x$ is rational and $f(x) = 1 - x$ when $x$ is irrational is continuous at $x = \frac{1}{2}$  To prove the continuity it is necessary to prove that for every $\epsilon$ there exists a $\delta$ such that $|f(x) - f(\frac{1}{2})| &lt; \epsilon$ whenever $|x -c| &lt; \delta$ $- (1)$ Now if we choose $a  € (\frac{1}{2} - \delta, \frac{1}{2} + \delta)$ and also $a $ is rational and choose $\epsilon &lt; |a - \frac{1}{2}|$ then condition in $(1)$ is not satisfied. How to move from here? I want to install the PyCharm IDE by JetBrains for Python (free community edition).  I managed to , but I do not know how to install it. Can you please help me? Galois extension of intersection of fields I have finite Galois extensions: E/K and E/L. $$M:=K \cap L$$ I am trying to prove that if the extension E/M is finite then it is also Galois. Any suggestions? Thanks What causes the 'Connection Refused' message? This is a  about Connection Refused    We see a lot of questions to the effect      When I try to connect to a system I get a message      Connection refused      Why is this ? Automorphism of rational numbers Let $\mathbb Q$ be group of rational numbers under addition.  Then I want to find out the automorphism group of $\mathbb Q$. In a book it is given that Aut($\mathbb Q$) is isomorphic to $\mathbb Q^*$, the  multiplicative group of rational numbers. I stuck in this. Google's CDN is nice, but it's not 100% reliable for all users. It occasionally fails to load for me. Other users , and some .  It's one more possible point of failure. It would be preferable if Stack Exchange didn't depend on it.  If you don't want to stop using it, consider adding a fallback if it fails to load. Something like this:  &lt;script src="https://ajax.googleapis.com/ajax/libs/jquery/1.7.1/jquery.min.js"&gt;&lt;/script&gt; &lt;script&gt;   if (!window.jQuery) {     document.write('&lt;script src="https://cdn.sstatic.net/stackoverflow/content/js/jquery-1.7.1.min.js"&gt;&lt;\/script&gt;');   } &lt;/script&gt; Let $A \in {M_n}$ is nonsingular and each eigenvalue of $A$ is either $+1$ or $-1$.Why $A$ is similar to ${A^{ - 1}}$? How to make a surface with a transparent background in pygame How does the Southwest Airlines boarding process work? I'm flying this week on Southwest Airlines for the first time. I paid for "Early Bird" automatic checkin and the FAQ mentions something about "A" boarding passes. I have no idea what that means, and I know that Southwest doesn't have assigned seats.  How does the boarding process work and is there anything I should be aware of? I have two laptops: Dell xps M1210 and Dell vostro 1400. On MS windows the output volume is so loud that I just need to set volume to around 20% to listen music and watch videos. I use Bose TriPort In-Ear Headphones most of the times. On ubuntu, even if I set 100% volume on the multimedia player, on the desktop volume manager as well, and also on the ncurses GUI of alsa mixer, I get very low volume output. Sometimes while playing some vidoes the volume is so low that I barely hear any words. I have struggled for two years since I started to use ubuntu to solve this mystery. The same sound card which gives excellent sound on MS windows fails miserably on GNU/Linux. Few weeks back I bought an Asus 100PX EEE PC. To my surprise even it also has the same issue. I have tried many versions of ubuntu and also I have done manual compiles of alsa modules without any success. I have asked and read about it in every possible forums and howto's. Nothing has helped me to address the issue. I have the following browsers on my phone:   stock Dolphin Mini Opera Mobile Firefox   I don't want to set a catch-all default because some pages render better on some browsers than on others.  When I click on a link on a non-browser app (e.g. google search or google reader) it asks me which browser to choose (as expected).  Can I set a default browser for some pages and a different default for others? I'm considering buying the MacBook Pro (Late 2016) with Touch Bar. As I'm a programmer and a Ubuntu user, could anyone tell me if the Touch Bar is compatible with Ubuntu? Why is the multiplicative group of a finite field cyclic? What is the measure of the set of numbers in $[0,1]$ whose decimal expansions do not contain $5$? Change edit approval/rejection I accidentally clicked reject on the suggested edit on this question(). I can't find the button to change it to Approve? How can I reduce screen brightness below the minimum without an app? I have an Optimus Elite, and it has a minimum brightness setting. In certain apps, I can reduce the brightness below the safe minimum, but when I leave the app, the brightness is increased to the system-wide minimum.  Using , I can permanently set the brightness below the safe amount, and it will stay even after going to other apps. Or using , I can accomplish almost the same thing even without needing root (it displays a translucent 'filter' over the whole screen, reducing brightness/contrast but not actually changing the backlighting power amount).  What I want to know is where the minimum screen brightness is stored. I have a rooted phone with CyanogenMod 9, so I have permissions to edit anything and could change a config or sqlite entry, but I don't know where this value is stored. What I would like to do is use the status bar brightness slider to be able to slide to the actual minimum of the screen, rather than the OS-controlled minimum. Having an app is useful, but having the functionality without the app seems to be within reach, if only I knew where the OS is getting it's safe minimum setting.  Where is this "safe minimum brightness" setting stored? If I can change it, I can remove my apps and use native controls. How to install Windows 8 without the setup DVD that came with my notebook? Currently look after 3 websites. I changed one of them to be HTTPS as it's ecommerce, however have seen a huge drop in organic traffic since this was implemented. Went from around 800 impressions to 400.   Any ideas as to why?   Very concerning as I expected it to improve if anything!  I set up 301 redirects to carry through to old pages to http://www.example.com/product1 would go to https://www.example.com/product1  I also set up a separate webmaster account for the HTTPs version as our integration wasn't working well with Google Merchant Centre, and data is flowing correctly but unsure why/if this would cause conflict.  Very confused!