Say a function f is differentiable. Prove whether its derivative is continuous over its domain or not.  On an intuitive level, I believe it should be always continuous over the domain (open intervals) and am thinking of a rigorous proof.
If a function $f$ defined on $[a,b]$ and differentiable everywhere, does it mean that its derivative $f'(x)$ is continuous everywhere on $[a,b]$?  My understanding is 'yes'. Because if there were 'a gap' in $f'(x)$, we could integrate back to $f(x)$ and show that where the gap is, there would be a sharp "angle" where $f(x)$ is not differentiable.
Possible Duplicate:          I was wondering if there's any good reason to still use blob fields in a database. A couple of years ago I worked with a DB with a bunch of images in it, the DB was very slow and I couldn't see any good reason to keep the images inside the DB, so I got the images out and stored the filenames instead.  Was this a smart move? What would you do in my place? Long rectangular quads? Or neat squarish quads with tris? For curved meshes I've often noticed that in my desperation to avoid tris, I tend to extrude the mesh in a way which creates these long rectangular quads:    In this case at least, the mesh is all quads, however the edge flow causes loopcutting to be awkward in some places:    I tried creating the same mesh with more even topology, however I found tris unavoidable. Maybe it was because I was starting with the long-quad mesh, but I also found myself using the knife tool for everything..    This makes loopcuts less necessary and easier to place in most cases, though the occasional tri does get in the way.  An annoying thing about the tris are the poles, though they aren't too noticeable in this particular case.    Both techniques seems to have their advantages and their disadvantages:    Which is considered "better", if any? Are there any better workflows for creating even topology with as few tris as possible? Or is there some trick I'm missing altogether? Simplest way to send one-line mail out via command line using gmail? Using Raspbian and Ubunntu 16.04 LTS so need a generic Linux solution.  Requirement is simple:  I need a way to send one-line email messages from the command line.  I have set up a gmail account just for this particular Rpi3, with the address of rpi3abc@gmail.com - with no 2FA  So now I need to be able to send one-line mail messages from anywhere (including cron) without user intervention.    I also would like it to be able to send text files; basically, anything from stdin. Show that every prime $p>3$ is either of the form $6n+1$ or of the form $6n+5$ I am a US Green card holder with an Indian passport expiring in 4 months planning a short weekend trip by road to Canada next month.  Will I be allowed to enter Canada considering my passport won't meet the "valid passport" criteria of 6 months? Can I achieve a torchlight effect (lighter area around a light source) in a 2D game? I am thinking of writing myself a simple 2D game. It will not shine with perfect graphics or gameplay at first, but I'd consider it my first step in PC game development. So, imagine such simple sprite-based 2D game (like Heroes IV or Startcraft BroodWar).  I want the gameplay to support day/night with the according lighting changes and at the same time it will be madness to have to create sprites for every lighting nuance. So, I decided adding a semi-transparent layer on top of other objects will be enough.   The issue with this solution is if I have a light source object in the game (like the hero wearing a torch, or a burning building), there must be a lighter area around it, right? Since I am putting my semi-transparent layer over everything, how would you suggest to achive the torchligt visual effect I want? Maybe redraw that layer adding 'gaps' or differently colored areas based on the lighting effect? Extra Backslash Required in First Appended Newline in sed Suppose $G$ is a finer filter than $F$ in a topological space $X$.  Is the net base in $G$ a subnet of the net base in $F$?  I am using the definitions of General Topology of Willard:     Definition 12.15. If $(x_\lambda)$ is a net in $X$, the filter generated by the filter base $\mathscr C$ consisting of the sets $B_{\lambda_0}=\{x_\lambda; \lambda\ge\lambda_0\}$, $\lambda_0\in\Lambda$, is called the filter generated by $(x_\lambda)$.      Definition 12.16. If $\mathscr F$ is a filter on $X$, let $\Lambda_{\mathscr F}=\{(x,F); x\in F\in\mathscr F\}$.   Then $\Lambda_{\mathscr F}$ is directed by the relation $(x_1,F_1)\le(x_2,F_2)$ iff $F_2\subset F_1$, so the map   $P\colon \Lambda_{\mathscr F} \to X$ defined by $P(x,F)=x$ is a net in $X$. It is called the net based on $\mathscr F$.   The definition of subnet used in Willard is the second one from this post:   Thank you Numbered list in rows in tables I want to have numbered listing in each row, I've tried with \rownumber, but it doesn't work. I wrote a number at the beginning, but Could it be automatically?  Can you help me? This is my code:  \documentclass[10pt, a4paper]{article}  \usepackage{fontspec} \setmainfont{Arial}  \usepackage{color, colortbl} \definecolor{Gray}{gray}{0.9}  \begin{document}  \begin{table}[h!] \tiny \begin{tabular}{|&gt;{\raggedright\arraybackslash}m{30mm}|&gt; {\raggedright\arraybackslash}m{5mm}|&gt;{\raggedright\arraybackslash}m{5mm}|&gt;{\raggedright\arraybackslash}m{5mm}|&gt;{\raggedright\arraybackslash}m{5mm}|&gt;{\raggedright\arraybackslash}m{5mm}|&gt;{\raggedright\arraybackslash}m{5mm}|&gt;{\raggedright\arraybackslash}m{5mm}|&gt;{\raggedright\arraybackslash}m{5mm}|&gt;{\raggedright\arraybackslash}m{5mm}|&gt;{\raggedright\arraybackslash}m{5mm}|&gt;{\raggedright\arraybackslash}m{5mm}|&gt;{\raggedright\arraybackslash}m{5mm}|} \hline \multicolumn{1}{|&gt;{\centering\arraybackslash}m{30mm}|}{\rowcolor{Gray}\textbf{Text}} &amp; \multicolumn{1}{|&gt;{\centering\arraybackslash}m{5mm}|}{\textbf{Sem 1}} &amp; \multicolumn{1}{|&gt;{\centering\arraybackslash}m{5mm}|}{\textbf{Sem 2}} &amp; \multicolumn{1}{|&gt;{\centering\arraybackslash}m{5mm}|}{\textbf{Sem 3}} &amp; \multicolumn{1}{|&gt;{\centering\arraybackslash}m{5mm}|}{\textbf{Sem 4}} &amp; \multicolumn{1}{|&gt;{\centering\arraybackslash}m{5mm}|}{\textbf{Sem 5}} &amp; \multicolumn{1}{|&gt;{\centering\arraybackslash}m{5mm}|}{\textbf{Sem 6}} &amp; \multicolumn{1}{|&gt;{\centering\arraybackslash}m{5mm}|}{\textbf{Sem 7}} &amp; \multicolumn{1}{|&gt;{\centering\arraybackslash}m{5mm}|}{\textbf{Sem 8}} &amp; \multicolumn{1}{|&gt;{\centering\arraybackslash}m{5mm}|}{\textbf{Sem 9}} &amp; \multicolumn{1}{|&gt;{\centering\arraybackslash}m{5mm}|}{\textbf{Sem 10}} &amp; \multicolumn{1}{|&gt;{\centering\arraybackslash}m{5mm}|}{\textbf{Sem 11}} &amp; \multicolumn{1}{|&gt;{\centering\arraybackslash}m{5mm}|}{\textbf{Sem 12}}\\ \hline 1. Text text text text text text text text text text text text text &amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\\\hline 2. Text text text text text text text text text text text text text &amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\\\hline 3. Text text text text text text text text text text text text text  &amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\\\hline 4. Text text text text text text text text text text text text text &amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\\\hline  \hline \end{tabular} \end{table}  \end{document} Would it be considered cheating to ask for homework help on the internet, if I openly acknowledge the help received? Note: If you're looking for a simple explanation as to why comments aren't required on downvotes, see .   I used to get "upset" (though that is too strong a term) when I got downvoted without comment. If my answer isn't good enough then I'd like to know why. Not only does it improve the answer for the OP but it improves my knowledge too.  Where the down-vote has been explained I've found it useful &amp; it has improved my answer, or forced me to delete the answer if it was totally wrong.  So is there any way we can encourage people to leave a comment? Perhaps they don't lose rep if they explain their down-vote?  I must admit though that I haven't always explained my down-vote either so you could call me a hypocrite. I've also grown a thicker skin over the months of using SO (it seems to have come with the higher rep score ;-)), so I'm less bothered about this now. There is no continuous injective function $f:\mathbb{R}^n\longrightarrow\mathbb{R}^m$ I have the following question, more general that   Let $U\subset\mathbb{R}^n$ be a domain. If $f:U\longrightarrow\mathbb{R}^m$  ($m&lt; n$) is a continuous function, how to prove that $f$ is not injective.  Any hint would be appreciated. How can I stand on the ground? EM or/and Pauli? There is this famous example about the order difference between gravitational force and EM force. All the gravitational force of Earth is just countered by the electromagnetic force between the electrons on the surface of my feet and the ground.  But also wave-function of electrons(fermions) do not overlap each other due to Pauli exclusion principle.  So which one is the true reason of me not flopping inside the earth forever? Is it only one of them(my guess is Pauli exclusion) or is it both? How do I summon mobs with block heads on Minecraft? I am developing a set of classes that implement a common interface. A consumer of my library shall expect each of these classes to implement a certain set of static functions. Is there anyway that I can decorate these class so that the compiler will catch the case where one of the functions is not implemented.  I know it will eventually be caught when building the consuming code. And I also know how to get around this problem using a kind of factory class.  Just curious to know if there is any syntax/attributes out there for requiring static functions on a class.  Ed Removed the word 'interface' to avoid confusion. Is there a reason to use multiple question marks instead of an asterisk in filename expansion? If I have a number of directories, named, for example 10001 through 10025 is there any reason to use ls 1*/foo vs. ls 100??/foo?  I have a lot more than 25 of them, so I mostly curious if there's any differences in speed.  I know the difference in use between the two, that the asterisk will match longer file names, like 10001.backup. But let's say I don't have any files that don't follow my conventions. Is there any behind-the-scenes differences? Surjectivity of Composition of Surjective Functions Suppose we have two functions, $f:X\rightarrow Y$ and $g:Y\rightarrow Z$. If both of these functions are onto, how can we show that $g\circ f:X\rightarrow Z$ is also onto? Not enough reputation - wanting to answer/comment I've seen a few questions I'd like to comment on or give a different answer, but seeing how I don't have 50 reputation (I've been a lurker for years, finally decided to participate) do I really need to ask a few questions before I can comment/answer other people's stuff? Or am I missing something here?  Sorry if this has been answered before, but with a few search tries I couldn't find any similar stuff, even tried google just for "50 reputation stackoverflow" but that didn't give me anything either. Issue when promoting a second server as domain on R2 2008 I'm in an odd situation here.  I've upgraded my network to a 2008 domain from a 2003 with out any issue, this works fine and all the FSMO are in one DC,  I was about to install a second domain on a 2008R2 and this error is coming up again and again.  I The problem is I keep getting an area telling me that I need to run adprep /domainPrep /forestPrep, but when I run it on the first DC, it tells me this has already been run and updated and it is aborted. Does anyone have any ideas on why I can't add a Server 2008 machine as a domain controller? How compute $\lim_{p\rightarrow 0} \|f\|_p$ in a probability space?