Cannot set int? to null inside of ?: Operator
Nullable type issue with ?: Conditional Operator
Show that function $f$ has a continuous extension to $[a,b]$ iff $f$ is uniformly continuous on $(a,b)$ Let $E \subset F \subset X$ and $f:E\rightarrow Y$. We say that the function $g:F\rightarrow Y$ is an extension of $f$ if $g(x) = f(x)$ for all $x \in E$.   Let $f: (a, b) \rightarrow \mathbb{R}$. Show that $f$ has a continuous extension to $[a, b]$ if and only if $f$ is uniformly continuous on $(a, b)$.   $(X,d)$ is a metric space.  Hint: To prove $\Leftarrow$ start by showing that $f$ maps Cauchy sequences to Cauchy sequences. Then how should you define $g(a)$ and $g(b)$?  The $\Rightarrow$ part of the proof was quite simple, but even with the hint, the other way seems a bit difficult. Doesn't the first part of the "hint" follow from the way we defined $f$? If so, how do I prove this? Help/Hints appreciated! Proving a trig identity: $\frac{\sin(A + B)}{\sin(A - B)}=\frac{\tan A + \tan B}{\tan A - \tan B}$ In the Avengers movie, why is Loki sweating and weak when he first arrives on Earth, in the SHIELD base? Can I be admitted into a PhD program with non-thesis MS degree? If I do a non-thesis MS program in statistics, will I be eligible for an admission in to a PhD program in Statistics later in my life? The MS program I am looking into is a non-thesis program but I still have to complete a research project requirement (which involves oral presentation as well as some form of a written report) towards the end of the program...I am still not sure if it can lead to a PhD program though.  Thanks! Open-source alternative to Adobe InDesign There is some open-source software, similar to Adobe InDesign? It would be awesome if it do exist...  I know all the others like Inkscape, Gimp, etc... but for InDesign i have none.  Could you give me one direction? Assume you have already completed your undergrad and have been working for a number of years. Does one need to be part of a university or a learning institution to publish papers? Prove that for any finite sequence of decimal digits, there exists an $n$ such that the decimal expansion of $2^n$ begins with these digits. Measures: Atom Definitions Let $\Omega$ be a measure space with measure $\mu$. (Here, a measure is only meant to be countable additive!)  Consider a subset $A\in\Sigma$.  Then according to the  it is an atom if: $$(1)\quad\forall E\subseteq A:\quad\mu(E)&lt;\mu(A)\implies\mu(E)=0\quad(\mu(A)&gt;0)$$ and according to the  it is an atom if: $$(2)\quad\forall E\in\Sigma:\quad\mu(E\cap A)=0\lor\mu(E^c\cap A)=0\quad(\mu(A)&gt;0)$$  Now, these definitions agree for the really trivial cases: $$\mu\equiv0$$ $$\mu\equiv\infty$$ but they differ for the less trivial case: $$\quad\mu(E\neq\varnothing):=\infty,\,\mu(\varnothing):=0$$ namely the atoms are w.r.t. (1) all nonempty subsets whereas w.r.t. (2) only the singletons.  Excluding this pathological case, are the definitions equivalent? (More precisely, assume there exists a measurable subset $0&lt;\mu(F)&lt;\infty$.)  So far I checked that: $$(2)\implies(1):\quad\mu(E)&lt;\mu(A)\implies\mu(A\setminus E)&gt;0\implies\mu(E)=0$$ but what about the other direction? How can you describe someone who gives up too easily? I'm trying to define \conjunction by means of \textbullet instead of \bullet, since the latter looks too big in logical formulas.  How can I make the spacing around \conjunction to behave like the spacing around \bullet when there is an opening parenthesis just after it?  Compare (2) and (2'):     \documentclass[12pt]{article} \usepackage[paperwidth=10cm,paperheight=7cm,margin=1em]{geometry} \usepackage{amsmath} \usepackage{amssymb} \usepackage{ebgaramond-maths}  \DeclareMathOperator\conjunction{\raisebox{-0.25ex}{\textbullet}}  \begin{document}\begin{align} \intertext {Good spacing but bad size:} (\exists x(\forall y(\phi(y)\equiv y\!=\!x)&amp;\bullet Gx)\supset\psi)\\ \exists  x(\forall y(\phi(y)\equiv y\!=\!x)&amp;\bullet (Gx\supset\psi))\\ \intertext {Good size and spacing:} (\exists x(\forall y(\phi(y)\equiv y\!=\!x)&amp;\conjunction Gx)\supset\psi)\\ \intertext {Good size but bad spacing:} \exists  x(\forall y(\phi(y)\equiv y\!=\!x)&amp;\conjunction (Gx\supset\psi)) \end{align}\end{document}   This won't do:  \DeclareMathOperator\conjunction{\raisebox{-0.25ex}{\textbullet\ }}   because it adds an unwanted extra space to (1'). What is the reasoning behind the design decision to allow users to create a display name which matches another existing users exactly?    This is opposed to the common practice of enforcing unique user names, and I'd really like to learn something. I have a DEM layer from ASTER satellite and I want to get the slope information from it.  I use terrain analysis > slope, but the output is so wrong. Why is that?  Here is the DEM:     And here is the "Slope" that is totally wrong. It should stretch the values from 0 - 89.9097 but it does not. What's the name for the relativistic paradox with the train/car travelling over a gap? There is a relativistic paradox I was introduced by a friend in Ithaca (Courtenay Footman). It involves a vehicle travelling over a ravine relativistically, so it's in the general family of ladder paradoxes. There are many ways to phrase it, but here's a simple one.   A three-car train with each car of length $L$ is travelling over a bridge across a ravine. (Assume that the couplings between cars are stiff: this train does not bend.) Unfortunately, some evil genius has precisely removed a length $6L$ of the track. The train however is relativistic, travelling at a speed such that $\gamma \approx 12$.  In the reference frame of the ground, the train should appear to be 1/24th the size of the gap, which suggests that it will fall a little and should crash into the bridge opposite. But in the reference frame of the train, each car of the train is at least twice as long as the gap and aside from the very front wheel and the very back wheel, which need to be supported by rigidity, everything is constantly well-supported. This suggests that the train makes it. Who is right?  I vaguely remember that as an undergraduate knowing only a little relativity, I trusted the train's perspective because it's at rest in that perspective, and Courtenay said that this was ultimately correct after a lot of details got resolved. But I do not remember all of these details such that I could work it out for myself. So I'm looking for some discussion that's a little more well-sourced or authoritative.  What is this paradox called, and what literature has addressed the issue? What is the chance that at least two people were born on the same day   of the week if there are 3 people in the room?   I'm wondering if my solution is accurate, as my answer was different than the solution I found:  Probability that there are at least 2 people in the room born on the same day = 1 - (No one was born on the same day) - (Exactly one person was born on the same day)  There are (3 choose 2) different pairs of couples. Each couple has the same birthday as another couple with the chances of 1/7 and different with chances 6/7. Thus:  $$1 – (6/7)^3 – 3(1/7)(6/7)^2 = 0.0553$$  Thanks for any help! Suppose you had a question, and you really want a good answer that would solve your problem, but noticed it was already asked by someone else. However, no one has posted any answers. The question is technically considered "unanswered", but a lot of users who would otherwise bother to give a correct answer probably skip the question completely for some reason.  What should I do in this case?  One idea is to ask a new, completely identical question, in the hope users will notice the new question and post an answer there. But would that go against SE policy?     Note: This is adapted from , but that doesn't cover the case where the original question is unanswered. Also, it is not a duplicate of , as that question asks about getting attention for one's own questions, while this one and the one linked above ask about getting an answer if someone else has already posted the same thing. When should I use "didn't" instead of "haven't"? Is haven't supposed to mean something I may still do and didn't not?  Can I use didn't for something I might still do? How can I remove malicious spyware, malware, adware, viruses, trojans or rootkits from my PC? Writing to NTFS drive in Linux I've got an external USB drive that is formatted with NTFS mounted on ~/tmp.  I can read it just fine, but cannot write to it, even as root (temp is a folder on the USB drive):  [root@Plugbox temp]# touch test touch: cannot touch `test': Permission denied [root@Plugbox temp]#   However, if I do a cat /proc/mounts, it looks like I do have NTFS read/write support:  [root@Plugbox temp]# grep /dev/sdb1 /proc/mounts /dev/sdb1 /root/tmp ntfs rw,relatime,uid=0,gid=100,umask=02,nls=utf8,errors=continue,mft_zone_multiplier=1 0 0 [root@Plugbox temp]#   Finally, if I go to the folder above temp, it looks like I should have plenty of permissions to write to the folder:  [root@Plugbox tmp]# ls -al total 61500764 ... drwxrwxr-x 1 root users           0 Apr 19  2010 temp ... [root@Plugbox tmp]#   What am I missing? This occurs in the  released recently to the public. When someone posts a code block in their question or answer, the code block is cut off on the right hand side, and I can't scroll to the right to view the rest of it. How can I view these code blocks? How to install with Wubi when it says "Could not retrieve the required disk image files"?