Showing Continuity of a curve Let $f(t)=t \sin \frac 1 t$ if $0 &lt; t \le 1$, and $f(t)=0$ if $t = 0$.  Consider the curve $\gamma$ with parametric representation $\phi : [0, 1] → \mathbb{R}^2$ defined by $\phi(t) = (t,f (t))$. Prove that $\gamma$ is a continuous curve which is not rectifiable.  for continuity, do I just show that $t\sin\frac{1}{t}$ is continuous on the interval given? To prove it is not rectifiable, I have to show that length of $\phi$ is infinite. Do I show the integral of the derivative of $t \sin \frac 1 t$ is infinite?
Rectifiability of a curve Let $f$ be a function defined on $[0,1]$ by   $$f(x) = { 0, \text{ if } x = 0} $$ $$f(x) = { x \sin \frac 1 x , \text{ if } 0 &lt; x \leq 1} $$  Prove that the curve $\{(x, f(x)) : x \in [0,1]\}$ is not rectifiable.   I'm not sure how to approach this. The general idea seems logical, we're proving that the length of the curve is infinite, but the method seems difficult to find.
A water pipe from sea level to beyond the atmosphere If a pipe extended from just above the ocean floor to outside the atmosphere, would water be sucked up it by the vacuum beyond the atmosphere? If a hole was made in the pipe, above sea level, how would that affect the flow of water? Would it stop it completely? How to make sure an application keeps running on Linux I'm trying to ensure a script remains running on a development server. It collates stats and provides a web service so it's supposed to persist, yet a few times a day, it dies off for unknown reasons. When we notice we just launch it again, but it's a pain in the rear and some users don't have permission (or the knowhow) to launch it up.  The programmer in me wants to spend a few hours getting to the bottom of the problem but the busy person in me thinks there must be an easy way to detect if an app is not running, and launch it again.  I know I could cron-script ps through grep:  ps -A | grep appname   But again, that's another hour of my life wasted on doing something that must already exist... Is there not a pre-made app that I can pass an executable (optionally with arguments) and that will keep a process running indefinitely?  In case it makes any difference, it's Ubuntu. Is there a way to recognize destructible environment in XCOM:Enemy Unknown? Two different solutions to integral How to implement a progress bar using the MVVM pattern A folder was created in my home folder (not by anyone), and I cannot see the folder contents. She has root privileges, but when the nautilus root access the folder is empty. When I try to remove it, it will appear to remove more than 1 million files from my computer, and I do not know what these files are.  How i can see these files? the name of folder created is "4F8ZBREShQ". I'm scared about it... How can I have 2 different mod folders with the new Minecraft launcher? Equivalent of "both" when referring to three or more items? What is the difference between the | and || or operators? I have always used || (two pipes) in OR expressions, both in C# and PHP. Occasionally I see a single pipe used: |. What is the difference between those two usages? Are there any caveats when using one over the other or are they interchangeable? Something very odd is happening here.   I have a query that looks like this.  SELECT CAST(FT.DOP AS SMALLINT) FROM TRACKING_DATA WHERE date &gt; @mydate and identifier = 0000000000   When run as a raw query it returns data just fine.  When i place it in a stored procedure that alters the where clause it throws this error.  Msg 244, Level 16, State 2, Procedure myprocedure, Line 107 [Batch Start Line 2] The conversion of the varchar value '58629' overflowed an INT2 column. Use a larger integer column.   So here is the oddity. I go through all the possible data for that where clause with a query like this.  SELECT DISTINCT DOP FROM TRACKING_DATA where identifier = 000000000000   and  SELECT DISTINCT CAST(DOP AS smallint) FROM TRACKING_DATA where identifier = 000000000000   And this is what i get back.  17 12 9 19 8 14 6 16 11 13 7 10 0 18 5 15 4   Nowhere does it have anything remotely too large for a SMALLINT. So i thought, ok maybe its a non-printable ASCII character. But i can't find any.  I'm a little perplexed at the moment. It runs find as a raw query, explodes as a procedure and all possible data based on the where is valid. My only suspicion is that the query plan is doing something odd with filtering or maybe runs with different validation when run as a procedure. Creating checkbox parameter in Python script tool for ArcGIS Desktop? I am working on creating an ArcGIS tool from a Python script I am writing.  I am wondering if it is possible to have a checkbox parameter.    I want to have a parameter where the user selects a feature class, then from the feature class the user will choose the field for the upper most layer in their model, then I want the user to be able to choose what layers they want the script to run on with a checkbox structure derived from the upper most layer field.    Is this possible with python, and ArcGIS Desktop? Run true multiple process instances of gnome-terminal tmux: enable mouse only when shift pressed How to format a duration in java? (e.g format H:MM:SS) How to sandbox applications? I want to run small untrusted programs, but restrict them from accessing any files outside of their folder, network access, and everything else that they don't really need. What is the simplest way to achieve this? find $\sum_{k=2}^{\infty}\frac{1}{k^2-1}$ KeyListener not working How do I parse an ISO 8601-formatted date? I need to parse  strings like "2008-09-03T20:56:35.450686Z" into Python's datetime type.  I have found  in the Python standard library, but it is not very convenient.  What is the best way to do this? Pascal's triangle in tikz I would like to typeset the top part of Pascal's triangle. To get the triangle with the names of the binomial coefficients, i.e., {n \choose k}, I used the following code  \begin{tikzpicture} \foreach \n in {0,...,4} {   \foreach \k in {0,...,\n} {     \node at (\k-\n/2,-\n) {${\n \choose \k}$};   } } \end{tikzpicture}   The result is this   Now I want to be equally lazy and do something like this for the values of the binomial coefficients, i.e., replace {\n \choose \k} in the node label with \CalculateBinomialCoefficient{\n}{\k} where \CalculateBinomialCoefficient is a hypothetical macro that calculates the binomial coefficient. Has anyone done something like that?  The result should look like this: New global preferences: keyboard shortcuts and sticky topbar