Create a nested lists out of another nested lists with while loop
How do you split a list into evenly sized chunks?
Function always continuous in a Sobolev Space? Hy everybody got a quick question. I know that all function F in a Sobolev Space has a continuous representative called U such as U=F almost everywhere.  Lets take for example: The Sobolev space on ]-1,1[    the function F(x)=0 on x&lt;0 and F(x)=1 x>0  we can't find a continuous representative because it would be on a larger interval let say ]-eps ,eps[ so not in a Sobolev space the function F(x)=x on x&lt;0 and x>0 and F(0)=1 where we could find the representative U=x   My question is: if F isn't continuous is it sure that it isn't an element of a Sobolev Space? It was given was given as a necessary condition in my course.  Thanks for the help. I am looking for one word to describe someone who always does what he loves.   Loving means enjoying something very much, free-spirited means not restrained by convention. Is there a hybrid word?   I mean a word that describes someone who is independent and always does what he loves. Please also try to not make the word too esoteric. Also, the word needs to be positive.  How will it be used?  I am submitting an application for college, and they require me to supply a word to describe myself. I have always done what I love. So, I require a word to describe someone who loves what they want. There is no sentence, in particular, just a prompt asking me to describe myself in one word. how to sort by the day of the week? I have two files.  file1:  Dave 734.838.9800   Bob 313.123.4567   Carol 248.344.5576   Mary 313.449.1390   Ted 248.496.2204   Alice 616.556.4458      file2:  Bob Tuesday   Carol Monday   Ted Sunday    Alice Wednesday   Dave Thursday     Mary Saturday     I merged the two files.  file3 should look like this:  Name      On-Call     Phone   Carol     MONDAY      248.344.5576   Bob       TUESDAY     313.123.4567   Alice     WEDNESDAY   616.556.4458   Dave      THURSDAY    734.838.9800   Nobody    FRIDAY      634.296.3356   Mary      SATURDAY    313.449.1390   Ted       SUNDAY      248.496.2204     But I cannot get the weekdays to be in order. How do I go about doing that? g++ undefined reference with a shared library For each $y \in \mathbb{R}$ either no $x$ with $f(x) = y$ or two such values of $x$. Show that $f$ is discontinuous. This is from Apostol's Mathematical Analysis.  Let $f$ be a function defined in $[0, 1]$ such that for each real number $y$ either there is no value of $x \in [0, 1]$ such that $f(x) = y$ or there are exactly two such values of $x$. Prove that a) $f$ must be discontinuous on $[0, 1]$. b) $f$ has an infinite number of discontinuities in $[0, 1]$.  Long back I had tried to solve the problem. I reasoned for a) as follows: Suppose that $f$ is continuous and let $m, M$ be its minimum and maximum values on $[0, 1]$. Clearly if $m = M$ then $f$ is constant and clearly for every $x \in [0, 1]$ we have $f(x) = m = M$ which contradicts given property of $f$. So we must have $m &lt; M$. Clearly there are now four distinct points $a, b, c, d$ in $[0, 1]$ such that $f(a) = f(b) = m, f(c) = f(d) = M$. By checking possible linear orderings of $a, b, c, d$ I could find a find a value of $y \in (m, M)$ for which there exist more than 2 values of $x$ such that $f(x) = y$. However this approach looks very clumsy and I suspect much simpler and easier proof is possible. Part b) seemed bit complex and my only guess was that probably $f$ was discontinuous on each subinterval of $[0, 1]$ but I could not prove this claim (may be the claim is wrong!). Perhaps there is a simpler solution to this part of the problem. Apostol's also asks to provide an example of such a function. Any hints on the overall problem will be appreciated.  Update: After looking at Marc's answer, I think I have found a way to show that there are infinitely many discontinuities. Earlier the solution was a part of the question itself but it made the question look quite lengthy hence I moved it to a separate answer. Code: \documentclass[12pt]{article} \usepackage{pgfplots} \usepackage{float} \pgfplotsset{compat=1.17} \usepackage{tikz} \usetikzlibrary{shapes, arrows.meta, automata, positioning, matrix, calc} \usepackage[RPvoltages, american,siunitx]{circuitikz} \usepackage[margin=1in]{geometry}  \usepackage{steinmetz} \begin{document}         \begin{figure}[H]         \centering         \begin{circuitikz}             \draw(0,0) to[V = $\vec{V}_g$] ++(0, 3) to[R = 50 &lt;\ohm&gt;] ++(3, 0) to[C = 1&lt;\micro\farad&gt;, v = $\vec{V}_o$] ++(0, -3) to[short] ++(-3, 0);         \end{circuitikz}     \end{figure} \end{document}  Output:  As we can see, $\vec{V}_o$ is almost touching the capacitor. How to adjust the spacing for the voltage label? Actually, can  potentially solve the problem? Note that what I am trying to draw is based off of this: Making adult decisions "is" or "are" really not fun: Which is correct? Making adult decisions are/is really not fun.     What is the proper answer and why? My gut says "is" but I can't explain why. How to ignore the first line of data when processing CSV data? Make a momentary switch control a toggle What are the simplest, cheapest, smallest ways to make a momentary switch produce a 2-state toggling output (latching momentary switch)?  In other words, the output is continuously low, and when you momentarily press the button/tact switch, the output changes to continuously high, and then when you press it again, it switches back to low. I want to use Windows 7 or 8 but I don't know to install a software like VirtualBox which we have to do with the use of terminal. I don't want to use Windows as a main OS so I want to use it in VirtualBox. Can you tell me how to install downloaded files in Ubuntu. I once installed java runtime but don't know the meaning of the lines which I have written in terminal, I just did as the blog posted by a man says. Open local folder from link Let $X \ge 0$.  Show  \begin{align} \lim_{y\to\infty} y\,\Bbb{E}\left[ \frac1X \cdot1_{X&gt;y} \right] &amp;= 0 \quad\text{and} \\ \lim_{y \downarrow 0} y\,\Bbb{E}\left[ \frac1X \cdot1_{X&gt;y} \right] &amp;= 0. \end{align}  N.B.: The author doesn't assume $\Bbb{E}[1/X] &lt; \infty$.    My try:  Since the expectation appears on the RHS of Markov's inequality, I don't think this inequality can be applied to solve this question.  $$ \begin{aligned} y\,\Bbb{E}\left[ \frac1X \cdot1_{X&gt;y} \right] &amp;= \Bbb{E}\left[ \frac{y}{X} \cdot1_{X&gt;y} \right] \\ &amp;&lt; \Bbb{E}\left[ 1 \cdot1_{X&gt;y} \right] \\ &amp;= \Bbb{P}(X &gt; y) \\ &amp;\xrightarrow[y\to0]{} \Bbb{P}(X \ge 0) = 1 \end{aligned} \tag1 \label1 $$  For the first part $y \to \infty$, I apply Markov's inequality at the row $\Bbb{P}(X &gt; y)$ in \eqref{1}, so that $$\Bbb{P}(X &gt; y) \le \frac{\Bbb{E}[X]}{y} \xrightarrow[y\to\infty]{} 0?$$ But I don't have $\Bbb{E}[X] &lt; \infty$.  Thanks for any help. Why do the Jedi say "May the Force be with you"? Why does the Community user approve and reject edits? Why does the Community user sometimes approve or reject edits? How does a machine know if my edit is valid? What does it mean if my edit was rejected by Community?     See also: Linux / Bash, using ps -o to get process by specific name? Upgrade or replace a Wubi installation? I've got Windows 7 with Wubi 11.10 installed. I would like to upgrade to 12.04 but I'm not sure what the best path would be.  I'm not skilled enough to use Ubuntu all the time, and my iPhone requires me to run some Windows version for the damn iTunes software. I would love to run Windows XP but I can't figure out how to install it via USB -- my computer has no cd drive.  I'd like to run Windows XP and Wubi 12.04 on top of that. Or, perhaps, Ubuntu 12.04 natively and Windows XP alongside of that. So I guess I have 2 questions:   (lazy) Should I upgrade my Wubi from inside Ubuntu, or should I remove Wubi 11.10 from within Windows and install a brand new Wubi 12.04? (proper) How can I install Windows XP and Ubuntu 12.04 alongside each other?   Update: I am going to create a separate question post about the above #2 question... I realize it's wrong to ask two questions in one post when they are that different. How to set up RAID 1? Proofs involving $ (A\setminus B) \cup (A \cap B)$ Give short arguments for each part below to complete the reasoning. Please do not use Venn diagrams.     a. Prove that $A = (A\setminus B) \cup (A \cap B)$.   (Hint: You should prove that $(A\setminus B)\cup (A\cap B) \subseteq A$ and vice-versa)      b. Prove that $(A \setminus B) \cap (A \cap B) = \emptyset$   (Hint: Try a proof by contradiction for this statement.).    I'm not too sure exactly where to start on this. Both statements make sense logically, but I don't know how exactly to prove them. Can anyone point me in the right direction? I'm travelling from Madrid to Brisbane, but I have a 9 hours layover at Dubai Airport. I do not plan to leave the airport, and the connection is with the same airline (Emirates). I'm from El Salvador.    Do I need a transit visa? Sneak Attack has the same type , this is easy, unless there are multiple damage types.  A Flame Tongue does slashing (or piercing) damage, and fire. Can I chose the Sneak Attack damage in this case to be fire too?