Consider $C([a,b])$, the vector space of continuous functions in $[a,b] \subset \mathbb{R}$ to $\mathbb{R}$. Let be $f \in C([a,b])$, we define $\|\cdot\|_1$ as follows: \begin{equation*}  \|f\|_1 =  \int_{a}^{b}|f(x)|dx  \end{equation*} I must show that $(C([a,b]), \|\cdot\|_1)$ is not a Banach space. This is my answer the which one is wrong according to my teacher. Consider the following counterexample. Let $c =2^{-1}(a+b)$ y $f_n:[a,b] \rightarrow \mathbb{R} \hspace{.1cm} \forall n \in \mathbb{N}$ with the following association rule and graph:  It's clear that $\{f_n\}_{n \in \mathbb{N}} \subset C([a,b])$. We will show that $\{f_n\}_{n \in \mathbb{N}}$ it's a $||\cdot||_1-Cauchy$ sequence. Let be $\epsilon &gt; 0$ and $N = \displaystyle \left\lceil{ \frac{1}{2 \epsilon}}\right\rceil$, then: \begin{equation*}     ||f_n-f_m||_1 = \int_{a}^{b} |f_n(x)-f_m(x)|dx = \left| \frac{1}{2n}-\frac{1}{2m} \right| \leq \max \left\{ \frac{1}{2n}, \frac{1}{2m}\right\} &lt; \epsilon, \hspace{.1cm} \forall n,m \geq N \end{equation*} So the sequence is $||\cdot||_1-Cauchy$, however though this sequence converges to the following limit function, the limit function is clearly discontinuous. Let be $f:[a,b] \rightarrow \mathbb{R}$: \begin{equation*} f(x) =      \begin{cases}         1 &amp; \text{si $x \leq c$}\\         0 &amp; \text{si $x &gt; c$}     \end{cases} \end{equation*} Fixed $\epsilon_1 &gt; 0$ and $N_1 = \left\lceil{\displaystyle\frac{1}{2 \epsilon_1}}\right\rceil$, then: \begin{equation*} ||f_n-f||_1 = \displaystyle\int_{0}^{1} |f_n(x)-f(x)| = \frac{1}{2n} &lt; \epsilon, \hspace{.1cm} \forall n \geq N_1 \end{equation*} $\textbf{Now, the problem is that}$ (according to my teacher) I only have propose a discontinuous function wich is a limit of the sequence, this does not means that do not exits another continuous function wich is also the limit of the sequence. I understand the problem, but nothing comes up to me to solve this.
That's really embarrassing, however I need to ask it. I could not prove that the normed space    $\langle \mathscr{C} [0, 1], \| \cdot \|_1 \rangle $ is complete (as a metric space), where $\| f\|_1 = \int_0^1 |f(x)|dx$. I don't really know how to explicit a continuous functions as a limit of a Cauchy sequence. I tried to prove that the pointwise limit (of the Cauchy sequence) have just a finite number of discontinuities (I don't know if it's true) and then approximate it by a continuous function.  Thanks in advance.
What theorems can I use to prove this? ASP.NET MVC How to pass JSON object from View to Controller as Parameter I have a complex JSON object which is sent to the View without any issues (as shown below) but I cannot work out how Serialize this data back to a .NET object when it is passed back to the controller through an AJAX call. Details of the various parts are below.     var ObjectA = {         "Name": 1,         "Starting": new Date(1221644506800),          "Timeline": [             {                 "StartTime": new Date(1221644506800),                 "GoesFor": 200              }             ,             {                 "StartTime": new Date(1221644506800),                 "GoesFor": 100              }          ]     };   I am not sure how this object can be passed to a Controller Method, I have this method below where the Timelines object mirrors the above JS object using Properties.  public JsonResult Save(Timelines person)   The jQuery I am using is:          var encoded = $.toJSON(SessionSchedule);          $.ajax({             url: "/Timeline/Save",             type: "POST",             dataType: 'json',             data: encoded,             contentType: "application/json; charset=utf-8",             beforeSend: function() { $("#saveStatus").html("Saving").show(); },             success: function(result) {                 alert(result.Result);                 $("#saveStatus").html(result.Result).show();             }         });   I have seen this question which is similar, but not quite the same as I am not using a forms to manipulate the data.   I have also seen references to using a 'JsonFilter' to manually deserialize the JSON, but was wondering if there is a way to do it nativly though ASP.NET MVC? Or what are the best practices for passing data in this way? Spyder - touchpad scroll does not work I use the scroll function with my mouse and it works fine. But either in script, or in terminal sections of Spyder, I can't use the touchpad for scrolling. I do not experience any problems with my touchpad in other programs like Word, Chrome etc. Is there any option for this in Spyder, specific for touchpad?  I am using Lenovo Legion Y520, Windows 10. Adding a reference between Eclipse Java projects How to create a Geometric Polygonal Art portrait I've been interested in creating Geometric Polygonal art illustrations, i.e. making polygons and shapes with different color shading represent a face, or an object, see  set of .  How are they achieved? Is there some script for Illustrator which alters the image adding the shapes according to the colors or shades? How would I transform a normal image of Marilyn Monroe for example, to a polygonal illustration of her? listing files with pattern with starting and ending I want to list all files in a directory that start with 'abc' and end with '.zip'  I'm trying to use ls.  The directory has a lot of zip files starting with abc&lt;date&gt; and xvz&lt;date&gt;. I only want to get the list of the abc&lt;date&gt;.zip I have a simple spreadsheet, two columns "Amount" (col Q) and "Account" (col O).  I want to get the sum of the "Amount" column where the corresponding "Account" forum equals "Paypal".  This is as far as I can get!  =SUM(Q:Q)*(O:O="Paypal") I would like to determine the operating system of the host that my Java program is running programmatically (for example: I would like to be able to load different properties based on whether I am on a Windows or Unix platform). What is the safest way to do this with 100% reliability? Is there a label/goto in Python? Is there a goto or any equivalent in Python to be able to jump to a specific line of code? How to compute $\int_0^{\pi/2}\frac{\sin^3 t}{\sin^3 t+\cos^3 t}dt$? Calculating with Mathematica, one can have $$\int_0^{\pi/2}\frac{\sin^3 t}{\sin^3 t+\cos^3 t}\,\mathrm dt=\frac{\pi}{4}.$$   How can I get this formula by hand? Is there any simpler idea than using $u = \sin t$?     Is there a simple way to calculate $$ \int_0^{\pi/2}\frac{\sin^n t}{\sin^n t+\cos^n t}\,\mathrm dt $$ for $n&gt;3$? Could anyone come up with a reference for this exercise? How does editing work? What is an adjective referring to “someone who puts people down”? I was just thinking about someone I went to school with who had a penchant for putting people down, and I was trying to think of a word that describes this, but couldn’t find it. It would be a synonym of caustic, abrasive, discouraging, and undermining, but these all have slightly different connotations (to wear away", cause discouragement or to make fall). I want an adjective that has the connotation of a hammer hitting you on the head. The word does not necessarily need to describe someone who does this on purpose, out of evil intent. Is there such a word?    The word should give you a mental picture of someone barraging you with negative comments, but it not necessarily affecting you. Think of someone who so often has something negative to “contribute” that many can just either ignore it or roll their eyes at it, and that is the extent of its effect. I keep getting the mental picture of the word buffet as in, how waves buffet the shore. A constant hitting, without the hitting necessarily being devastating in its destruction. I was speaking to an English learner and said, “All you have to do is read a lot.” And they thought that sentence wasn’t grammatically correct because I dropped the word to between is and read.  They thought it should be “All you have to do is to read a lot.” That sounds weird to me.   How do I explain to them the reason you drop the to? Or am I incorrect and it really should have a to? How to recover data from a USB-connected hard disk that shows no partition? Prove that the polynomial $(x-1)(x-2)\cdots(x-n) + 1$, $ n\ge1 $, $ n\ne4 $ is irreducible over $\mathbb Z$ I try to solve this problem. I seems to come close to the end but I can't get the conclusion. Can someone help me complete my proof. Thanks          Show that the polynomial $h(x) = (x-1)(x-2)\cdots(x-n) + 1$ is irreducible over $\mathbb Z$ for all $n\ge1$ and $ n\ne4$.      Suppose $h(x) = f(x) g(x)$, then we must have $f(i)g(i) = 1$ for all $i = 1,2,...n$. So both $f(i)$ and $g(i)$ are $1$ or $-1$. In either case, $m(x) = f(x) - g(x)$ has degree smaller than $n$ and have $n$ different roots ($1,2,...,n$). So we must have $m(x) = 0$. Then $h(x) = f(x)^{2}$. So $n$ must be even. Let $n = 2k$. Because $f(x)$ has degree $k$, there are $k$ values from $\{1,2,...,2k\}$ at which $f(x)$ is $1$ and $k$ values at which $f(x)$ is $-1$.  This is where I got stuck. Hope some one can help me solve this. Thanks. How do I emphasize role-playing during combat? Right now, the group that I GM for is very, very tactical-combat focused. To some extent, we may as well be playing a dungeon crawl board game for all the role-play we manage.  Now, in and of itself that's not bad, and I enjoy progressing the story that we're telling, but I would like more "theatre of the mind" than we're achieving right now.  The one bit of advice that I always seem to come across is to have the villains and other NPCs in the fight invite role-play by taunting and otherwise engaging in banter with the PCs, but unfortunately I've tried and failed at that - largely because quick witted banter is not my forte, and so it comes across as strained and less than engaging.  What I would like are other play-proven tactics that I as a GM can employ to lead my players towards more role-play during combat, without taking them away from their (and my) roll-play comfort zone. How to deserialize a JSON String to Apex How can I deserialize this json object:      {     "response": {         "count": 1,         "benchmark": 0.22567009925842,         "requests": [             {                 "request": {                     "id": 537481,                     "image_thumbnail": "",                     "title": "Request for new bin(s) - residential",                     "description": "Propmain ref  3234-1114",                     "status": "submitted",                     "address": "36 Pine Tree Close",                     "location": "Peterborough, England",                     "zipcode": "PE1 1EJ",                     "user": "",                     "date_created": 1417173208,                     "count_comments": 0,                     "count_followers": 0,                     "count_supporters": 0,                     "lat": 52.599967,                     "lon": -0.233482,                     "user_follows": 0,                     "user_comments": 0,                     "user_request": 1,                     "rank": "0"                 }             }         ],         "status": {             "type": "success",             "message": "Success",             "code": 200,             "code_message": "Ok"         }     } }   What i've tried:   Map&lt;String,Object&gt; rawObj = (Map&lt;String,Object&gt;) JSON.deserializeUntyped(jsonString);     Map&lt;String,Object&gt; responseObj = (Map&lt;String,Object&gt;)rawObj.get('response');     List&lt;Object&gt; reqs = (List&lt;Object&gt;) responseObj.get('requests');     System.debug('Map Size = ' + reqs.size());     Map&lt;String, Object&gt; i = new Map&lt;String, Object&gt; ();     for (Object x : reqs) {          i = (Map&lt;String, Object&gt;)x; }     Map&lt;String,Object&gt; requests = (Map&lt;String,Object&gt;)i.get('request');     System.debug('Map Size = ' + i.size());     for (String field : i.keySet()){          Object id = i.get(field);         Object title = i.get('title');         System.debug('Id : ' + id);         System.debug('title : ' + title);          //System.debug('Title : ' + title);       } Passing variable arguments to another function that accepts a variable argument list File has different name in terminal and Finder I've just downloaded the driver for my preferred trackball from Logitech. In the Finder, the Installer's filename is Logitech Control Center Installer.app. In the Terminal, ls shows LCC Installer.app. How did this happen? Simulating data to fit a mediation model