The corollary says if $ f $ is differentiable on $ [a,b] $ then $ f' $ cannot have any simple discontinuities on $[a,b] $.   I just don't how to prove it.  I think it should be proved on both two cases of simple discontinuities(first type and second type of simple discontinuities).  Thanks in advance.
I have a question about the corollary to theorem 5.12 in Rudin's Principles of Mathematical Analysis (page 108):  Suppose $f$ is a real differentiable function on $[a,b]$ and suppose $f'(a)&lt; \lambda &lt; f'(b)$ then there is a point $x \in (a,b)$ such that $f'(x) = \lambda$ Corollary: If $f$ is differentiable on $[a,b]$ then $f'$ cannot have any simple discontinuities on $[a,b]$.  Can someone help me to show how he uses the result in the &quot;main theorem&quot; in the corollary?  (There are two cases of simple discontinuities  $f(x+) = f(x-) \neq f(x)$ and $f(x +) \neq f(x-)$
How to NOT show keyboard layout chooser popup when changing language in GNOME 3 Explicitly finding the sum of $\arctan(1/(n^2+n+1))$ When I open a window to air out the room, how does the smell disperse? Let's say I'm in a room with some kind of noxious stink, possibly of flatulent nature.  The quickest way to right the world that comes to mind is to open a window.  When I open a window, how do the stank particles leave the room? How do I migrate an SVN repository with history to a new Git repository? How can I reference the script element that loaded the javascript that is currently running?  Here's the situation. I have a "master" script being loaded high in the page, first thing under the HEAD tag.  &lt;!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"&gt; &lt;html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en"&gt; &lt;head&gt; &lt;script type="text/javascript" src="scripts.js"&gt;&lt;/script&gt;   There is a script in "scripts.js" which needs to be able to do on-demand loading of other scripts. The normal method doesn't quite work for me because I need to add new scripts without referencing the HEAD tag, because the HEAD element hasn't finished rendering:  document.getElementsByTagName('head')[0].appendChild(v);   What I want to do is reference the script element that loaded the current script so that I can then append my new dynamically loaded script tags into the DOM after it.  &lt;script type="text/javascript" src="scripts.js"&gt;&lt;/script&gt; loaded by scripts.js--&gt;&lt;script type="text/javascript" src="new_script1.js"&gt;&lt;/script&gt; loaded by scripts.js --&gt;&lt;script type="text/javascript" src="new_script2.js"&gt;&lt;/script&gt; When should I use velocity versus addForce when dealing with player objects? Where do I suggest new Stack Exchange sites? Is there an appropriate place to suggest new Stack Exchange based sites, perhaps a place for them to be voted upon?  I wish there was a Stack Exchange site for Aviation (both real-life and simulated). Two column float with [t] using the cas-dc template I am using the cas-dc template for Elsevier. When trying to use \begin{figure*}[t], except I get the error 'The module 'cas/fig' does not have a key called 'cas/fig/t'. Other templates I use work fine. I have included \usepackage{stfloats}. I don't see anything mentioned that strictly forbids it, unless I missed this. Anyone know why? Is an HTTPS query string secure? I am creating a secure web based API that uses HTTPS; however, if I allow the users to configure it (include sending password) using a query string will this also be secure or should I force it to be done via a POST? Can I use PCA (or should I use regression) for testing the effect of multiple variables on one dependent variable? I have 2000 soil property measures and 14 different variables like rainfall, temperature, slope, etc. I want to check the effect of those 14 variables on soil property measures, including which variable affects my soil property the most. Is PCA a suitable solution to this? Or should I try multiple regression instead? Can Sasuke use Tsukuyomi? Solving 2-degree equations in 3 variables. We are given 3 equations: $x^2+\sqrt3 xy + y^2 = 25$ $y^2 + z^2 = 9$ $x^2 +xz+ z^2 = 16$. $x,y,z$ are positive real numbers. Then we have to find value of $xy + 2yz + \sqrt3 xz$. Calculate the last digit of $3^{347}$ The units of $\mathbb Z[\sqrt{2}]$ How can I show that the units $u$ of $R=\mathbb Z[\sqrt{2}]$ with $u&gt;1$ are $(1+ \sqrt{2})^{n}$ ?   I have proved that the right ones are units because their module is one, and it is said to me to do it by induction on $b$ and multiplication by $-1+\sqrt{2}$. I have already shown that the units of this ring has norm $1$ and all the numbers with norm $1$ are units, this may help. Short story (50s? 60s?) with a man trapped inside a sentient cave Careers: What does "allows remote" mean? When a posting is marked "allows remote", does it mean work in the office some days, work from home some days?  Or does it mean you could actually live out of town and maybe fly in as needed?  The reason I ask is I'm really interested in public facing software rather than internal/enterprise stuff, and most of that seems to happen in a few very expensive cities.  Given it's so hard to hire in places like SV/NYC, it seems like it could be a good scenario for companies if a developer had a really solid list of accomplishments, but I don't know how many companies think this way. How do I disable System Integrity Protection (SIP) AKA "rootless" on macOs [OS X] Apple has introduced , also known as "rootless", with OS X 10.11, El Capitan. I understand this is a step for general protection against malware but as a developer I need write access to some of the files it locks away.  How do I disable this protection? Animated textures don't work in Blender viewport playback What's the purpose (if any) of "javascript:" in event handler tags? I've been making a concerted effort to improve my javascript skills lately by reading as much javascript code as I can. In doing this I've sometimes seen the javascript: prefix appended to the front of event handler attributes in HTML element tags. What's the purpose of this prefix? Basically, is there any appreciable difference between:  onchange="javascript: myFunction(this)"   and  onchange="myFunction(this)"   ? Could not download all repository indexes