I was drawing smooth curves through points $(a,f(a))$ and $(b,f(b))$ in an attempt to come up with a proof of the Mean Value Theorem. As I was doing that, I noticed that I couldn't draw the curve in such a way that its derivative would be discontinous. I tried to think up a function that is differentiable but whose derivative is discontinous but I got nothing. I did a quick Google search for the same question that I've asked here but all I could find was that differentiability of a function implies continuity of the same which I already know. If $f(x)$ is differentiable over $(a,b)$, does that mean $f'(x)$ is continuous over $(a,b)$? If not, what are some examples of functions where this is not the case?
Could someone give an example of a ‘very’ discontinuous derivative? I myself can only come up with examples where the derivative is discontinuous at only one point. I am assuming the function is real-valued and defined on a bounded interval.
How to install GUI desktop on a server? After installing Ubuntu server how do I install the desktop environment? Can't see a glass object through another glass object in Eevee I was just testing Eevee and found that you cannot view a glass object through another glass. Is there any workaround? It works fine only If I rotate the glass window or put my camera to the back side of the scene Is it okay to ask a professor from another university for class lecture notes of a course that he offered on that university? I find two very interesting courses that were offered in the Spring 2019 semester at ETH Zürich. Unfortunately, their class lecture notes are not shared publically. I wonder if it is appropriate to email the professors and ask them for class notes. Who is the Lord, as referred to in the title? Frodo? Sauron? Someone else? Or is it more of an abstract title, not referring to a single person, like "He who wields the One Ring"? A two column loop with one lead post I got a loop from internet (I forgot the site) for two column post view. It has a nice CSS for it too. But skipping that for the question now.  &lt;?php $counter = 1; ?&gt;                  &lt;?php query_posts( 'cat=3&amp;showposts=6' ); //Find whether the catID is 3 ?&gt;             &lt;?php while ( have_posts() ) : the_post() ?&gt;                   &lt;?php /* Two Column Support get class */ ?&gt;              &lt;?php $class = ( $counter % 2  ? ' one' : '' ) ?&gt;                  &lt;?php /* Two Column output open the div class */ ?&gt;              &lt;div class="column&lt;?php echo $class;// echo $first; ?&gt;"&gt;              &lt;?php news_format(); ?&gt;             &lt;/div&gt; &lt;!-- End Two Column Support close div class --&gt; &lt;?php $counter++; //Update counter ?&gt;   It printed my posts into 2 columns like:  ========= | 1 | 2 | --------- | 3 | 4 | --------- | 5 | 6 | =========  It's fine. But I want to print it like this:  ========= |   1   | --------- | 2 | 3 | --------- | 4 | 5 | --------- | 6 | 7 | =========  What the modification I need to do? I can't pass any custom limit into the WordPress' custom loop &lt;?php while ( have_posts() ) : the_post() ?&gt;. It's for a custom theme. I did it, but the problem is, it's showing like:  ========= |   1   | --------- | 1 | 2 | --------- | 3 | 4 | --------- | 5 | 6 | =========   is my whole bunch of code. What is the maximum or minimum tire width I can fit on my bicycle We often get questions on bicycles.stackexchange about fitting wider or narrower tires to bicycles. We have a large number of these with answers suiting the specific bike or style of bicycle in question, but I have been unable to find one that provides a generic answer covering off all things that need to be considered. For most bicycles, the answer is the same regardless of the specific style and sizes of tires in question.   This is intended as a canonical question that we can point closed questions to. Visual Studio - How to change the return value of a method in the debugger? When I'm debugging, I often have to deal with methods that does not use an intermediate variable to store the return value :     private int myMethod_1()    {       return 12;    }     private int myMethod_2()    {       return someCall( someValue );    }   Well, in order to recreate a bug, I often have to change values on the fly. Here, I could want to see what happens when myMethode_1 return zero. Same thing for myMethod_2.  Is there a way to do that without modifying the code ? What I would like to do, it's to place a breakpoint on the return line (or on the closing bracket) and type-in a new return value.  Note : I'm using Visual Studio 2005 and Visual Studio 2008.  Thank ! Valid US visa on expired passport. Can I travel to the US? My old passport is in my maiden name and has already expired. My B1/B2 visa to travel to United States is still valid but is stamped in my old passport with my maiden name.   My new passport has my new name (old name got changed after marriage) but has details (like old passport number, father and mother's name) related to my old passport.    Will I be allowed to travel to US as long as I carry the my old as well as new passport?  Or Do I need to again get the visa stamped on my new passport? How to close view windows? Gel circular polarising filter I am contemplating buying a lens that only takes rear gel filters. Is it possible to buy a circular polarising filter in gel format? If so, who can supply them (preferably UK based)? Why null cast a parameter? I tried the first experiment in chapter 5 in the Metafontbook: ran mf, typed \relax and drawdot (35,70); showit;.  This opened a window entitled Metafont, but containing nothing.  What's missing?  Using TeXlive 2020 on Linux. PS. After adding drawdot (65,70); showit; I see both dots. PS II. Which have now disappeared, after I changed window-manager workspace and back. PS III.  Typing showit; before the first drawdot opens a window and allows the first dot to be seen, but after having switched workspaces and back, I haven't found any way of bringing the picture back, short of drawing something more and showing it. how to fill color in image in particular area? Connecting to Oracle database with ArcGIS for Desktop? Uncountable well ordered set in $Z^-$ theory. I want to build an uncountable well-ordered set within the theory $Z^\textbf-$. So, I take $A=\omega$ (exists by infinity axiom) and define $$W:=\{(X,R)\in\mathcal{P}(A)\times\mathcal{P}(A\times A):\langle X,R\rangle\ \text{is a well-ordered set}\}$$  With this I consider the set $T:=W/\cong$ (where $\cong$ is isomorphism relation). Note that $W$ and $T$ are sets by power set axiom. So I define for each equivalence class $[x],[y]\in T$ the order  $$ [x]\leq_T[y]\Leftrightarrow \text{type}(x, R_x)\leq \text{type}(y,R_y) $$  Here, $R_x$ means the order of $x$ that make it belongs to $[x]$.   So, $T$ is well ordered by $\leq_T$ and it is uncountable.  Question: I don't pretty sure if I can build $\leq_ T$ without replacement axiom. Another doubt is, Can I take the $R_x$ without AC? How to install 7zip to extract rar files How do I install 7zip through Terminal to extract rar files? Can the Big Rip really rip apart an atomic nucleus? Some scenarios describing the fate of the matter vs  tug of war on the universe involve the acceleration of the universe increasing to the point that it ends up ripping apart even atoms. This is called the .  This makes no sense to me. It looks like all of these   (GR) models of the universe assume it has a uniform isotropic distribution of matter and energy. This works great at long scales, but it is also clearly wrong even at the length scale of the separation of galaxies.  The density of atomic nuclei remains the same even though the universe is expanding even as we speak. I don't think the "scale" in the  can be interpreted so literally. Or maybe said better, it's got to break down when the cosmological horizon distance gets to the scale where the isotropic assumption breaks down, right?  How can scientists be claiming that runaway expansion will rip apart atoms? How do I use Linq to obtain a unique list of properties from a list of objects? I'm trying to use Linq to return a list of ids given a list of objects where the id is a property. I'd like to be able to do this without looping through each object and pulling out the unique ids that I find.  I have a list of objects of type MyClass and one of the properties of this class is an ID.  public class MyClass {   public int ID { get; set; } }   What I want to do is write a Linq query to return me a list of those Ids  How do I do that given an IList&lt;MyClass&gt; such that it returns an IEnumerable&lt;int&gt; of the ids?  I'm sure it must be possible to do it in one or two lines using Linq rather than looping through each item in the MyClass list and adding the unique values into a list.  Any help in creating an elegant solution would be much appreciated! du not accounting for space shown by df I have a CentOS 7.2 (minimal) VM running in a vSphere cluster, which has a 16G drive formatted as XFS under the default full-disk usage partitioning scheme from the CentOS installer. The system is running a test httpd service (all software from the default centos repos), but its space usage on disk is growing. This is in part from problematic apache error/access logs (2+GB in size), and removing these longs frees up some space. What is eating the drive space isn't so much a concern for me (I'll just nuke and pave this eventually). However, when I run df an du to check file sizes, I get a discrepancy and this is what's confusing.  First I run df -h to view filesystem usage:  [root@svrhttp03 httpd]# df -h Filesystem                         Size  Used Avail Use% Mounted on /dev/mapper/centos_svrhttp03-root   14G  4.8G  9.1G  35% / devtmpfs                           902M     0  902M   0% /dev tmpfs                              912M     0  912M   0% /dev/shm tmpfs                              912M   97M  816M  11% /run tmpfs                              912M     0  912M   0% /sys/fs/cgroup /dev/sda1                          497M  169M  329M  34% /boot tmpfs                              183M     0  183M   0% /run/user/0   This is showing 4.8G used on the root FS.  I then run the following to view the size of all items at the root of the filesystem:  [root@svrhttp03 httpd]# du -a -h -t 10K / 2&gt; /dev/null | grep -v -E "[A-Z,a-z,0-9]/." | sort -h 16K /home 7.4M    /root 31M /etc 97M /run 143M    /boot 324M    /var 1.3G    /usr 1.9G    /   This shows only 1.9G in use, and the reported size of all directories add up to ~1.9G, as expected.  So how can I find that ~2.9G discrepancy between what df and du are claiming for data on disk? Can you help me with my software licensing issue?