I know that this result is true for uniform convergence, but I'm struggling to find a counter example of such function.
Give an example of a sequence of continuous functions $f_n:\mathbb{R} \to \mathbb{R}$ such that the pointwise limit $f:\mathbb{R} \to \mathbb{R}$ exists but is discontinuous. My idea is $f_n(x)=\displaystyle\frac{x}{n}.$ Then the pointwise limit is $0$ but $f$ is discontinuous on $\mathbb{R}.$ Is that correct? I'm having a hard time understanding the definition of pointwise convergence. I have this definition: $\displaystyle\lim_{n\to \infty} f_n=f$ pointwise iff $\displaystyle\lim_{n\to \infty} f_n(x)=f(x)$ But then what is $f(x)$? Where does the $n$ go? In the example I gave, for instance, $\displaystyle\lim_{n\to \infty} f_n(x)=\displaystyle\lim_{n\to \infty} \frac{x}{n}=0.$ But does this equal $f(x)$? Is $f(x)$=0 a discontinuous limit?
How iptables tables and chains are traversed I know linux has 3 built-in tables and each of them has its own chains as follow:  FILTER: PREROUTING, FORWARD, POSTROUTING  NAT: PREROUTING, INPUT, OUTPUT, POSTROUTING  MANGLE: PREROUTING, INPUT, FORWARD, OUTPUT, POSTROUTING  But I can't understand how they are traversed, in which order, if there is. For example, how are they traversed when:   I send a packet to a pc in my same local network when I send a packet to a pc in a different network when a gateway receives a packet and it has to forward it when I receive a packet destinated to me any other case (if any) Url decode UTF-8 in Python Ubuntu 19.04 and Gnome desktop icons Possible Duplicate:          On  it's a stated fact that if $I$ and $J$ are coprime ideals of a ring $A$, then $I^k$ and $J^\ell$ are also coprime for any integers $k,\ell$.   There's no proof given. Why is this fact true? Why does Stephen Hawking say black holes don't exist? Show this equality (The factorial as an alternate sum with binomial coefficients). Suppose we have $x^2+4x+3=0$  We differentiate both sides to get $2x+4=0$.  So $x=-2$  But $(-2)^2+4(-2)+3=0$   $4-8+3=-1$ is not equal to $0$.  Which tells that we cannot differentiate equalities. But in many proofs, like in integral calculus, differentiating both sides occurs. So, where is the fallacy? It turns out that removing a virtual package with apt-get only remove the package itself,  leaving behind the real packages that were installed by it  So how do you remove a virtual package, along with all the real packages that it installs? What's wrong with this derivation that $i\hbar = 0$? raw data stationary but still can see trend and seaonality is stl Why is there no article in "We are Family"? What is the promotion rate of professors in United States? I'm writing an essay, and want to know the promotion rate and average duration from assistant professor to associate professor and associate professor to professor in recent years. Sadly I don't know the data for the US. Could anyone tell me? How to add a loopback interface? I want to add two more loopback interfaces.  I used the following command:  ifconfig lo: 127.0.0.2 netmask 255.0.0.0 up   and type in ifconfig, i can see the loopback address was added.  But if i want to add one more interface... (for example 127.0.0.3) the previous interface (127.0.0.2) was overwritten.  Also when i look in /etc/network/interfaces i see no entry.  How can i add multiple loopback interfaces permanently? How to enable crash reports/core dumps/stack trace logging globally? Crashing bugs can be the most annoying ones, leading to data loss, downtime and frustrated users. It would be good if applications crashed less.  Because of the complexity of the machine context, crashes often can't be reproduced in reasonable time for an ordinary user. That doesn't mean that the bug is rare - It could simply mean that the thing that triggers it occurs rarely for each user (for example DST changes). Such bugs are unlikely to be fixed unless a lot of users report them. It would be good if more crashes were reported.  To debug crashes, the developers need as much unambiguous context as possible. Generated crash reports are good, because they're usually detailed and accurate. Users can't be expected to zealously observe and report all the context manually, so they often submit sparse and wrong information.  The target audience for a lot of applications is not developers or sysadmins, but rather the general public, at home or at work. Such users can't be expected to know how to collect crash information manually, or to install -dbg packages, but generated reports from such users can still be usable. Some applications have their own crash reporting tools, but in my experience these rarely work, and when they report that they failed to report the error there doesn't seem to be any information on how to do it manually (I've observed this for recent versions of both Firefox and Flash). System-wide generation of crash reports would be good.  Is there any sort of crash report generation* which can be enabled globally** without installing a ton of -dbg packages, reading the documentation of each and every application, or slowing a normal machine down to a crawl?  * Logs, core dumps, stack traces, whatever  ** Not necessarily for init, but at least for a significant subset of the applications running on a typical desktop Linux installation. In my experience GUI applications crash 100+ times more frequently than shell applications, so GUI applications would naturally be the focus. Don't heavier objects actually fall faster because they exert their own gravity? The common understanding is that, setting air resistance aside, all objects dropped to Earth fall at the same rate. This is often demonstrated through the thought experiment of cutting a large object in half. The halves clearly can't then fall more slowly just by being sliced in two.  However, I believe the answer is that when two objects fall together, attached or not, they do "fall" faster than an object of less mass alone does. This is because not only does the Earth accelerate the objects toward itself but the objects also accelerate the Earth toward themselves. Considering the formula: $$ F_{\text{g}} = \frac{G m_1 m_2}{d^2} $$  Given $F = ma$ thus $a = F/m$, we note that the mass of the small object doesn't seem to matter as when calculating acceleration the force is divided by the $m$ term, its mass. However, this overlooks that the force is actually applied to both objects, not just to the smaller one. The acceleration on the second, larger object is found by dividing $F$, in turn, by the larger object's mass. The two objects' acceleration vectors are exactly opposite, so closing acceleration is the sum of the two:  $$ a_{\text{closing}} = \frac{F}{m_1} + \frac{F}{m_2} $$  Since the Earth is extremely massive compared to everyday objects, the acceleration imparted on the object by the Earth will radically dominate the equation. As the Earth is $\sim 5.972 \times {10}^{24} \, \mathrm{kg} ,$ a falling object of $5.972 \, \mathrm{kg}$ (just over 13 pounds) would accelerate the Earth about $\frac{1}{{10}^{24}}$ as much, which is one part in a trillion trillion.  Of course in everyday situations, we can for all practical purposes treat objects as falling at the same rate because of this negligible difference—which our instruments probably couldn't even detect. But I'm hoping not for a discussion of practicality or what's measurable or observable, but what we think is actually happening.  Am I right or wrong?  What really clinched this for me was considering dropping a small Moon-massed object close to the Earth and a small Earth-massed object close to the Moon. This made me realize that falling isn't one object moving toward some fixed frame of reference, but that the Earth is just another object, and thus "falling" consists of multiple objects mutually attracting in space. Installing from source. How to resolve dependencies without destroying the package manager I was using the amspset document class for generating exam practice problems. I had wanted to add some referencing to the problems so I started a new counter like this (adapted from some Tex Exchange post)  \newcounter{P} \newcommand{\plb}[1]{\refstepcounter{P}\label{#1} \textbf{P-\ref{#1}.}}  \begin{problem} \plb{p1} A parallel-wire transmission line is constructed of \#6 AWG copper wire (diameter $d=0.162$ in., conductivity $\sigma_c=5.8\times 10^7$ S/m) with a 12-inch separation in air. Assuming no leakage between the two wires, find $R'$, $L'$, $G'$, and $C'$. Assume a working frequency of 1 MHz. \end{problem}  \begin{solution} Using Table.~\ref{tab:tlparam} and Fig.~\ref{fig:tlempl},  \[ Rs = \sqrt{\pi f \mu_c/\sigma_c} = 2.61\times10^{-4}~ \Omega\] \[ R' = \frac{2R_s}{\pi d} = 4.04\times10^{-2} ~\Omega\text{/m} \] \[ L' = \frac{\mu_0}{\pi} \ln{\frac{2D}{d}}=2.0~\mu \text{H/m} \] \[ C' = \frac{\pi \epsilon}{\displaystyle\ln{\frac{2D}{d}}}=5.56 ~\text{pF/m} \] $G'=0$ because the problem states that there is no leakage between the two wires. \end{solution}  \begin{figure}[ht] \begin{centering}     \includegraphics{fig24}     \caption{(Fig.2-4 from FAE) A few examples of transmission lines.} \end{centering} \label{fig:tlempl} \end{figure}  \begin{table}[ht] \caption{(Table 2-1 from FAE)Transmission-line parameters $R'$, $L'$, $G'$, and $C'$ for three types of lines.}     \begin{tabular}{c}         \hspace{3cm}\includegraphics{tb21}     \end{tabular} \label{tab:tlparam} \end{table} ... and more   Now all the figure references stop working, I get "Fig. " instead of "Fig. 1". The table references are alright though. Any idea what the problem could be? I made changes to iptables config file in /etc/iptables/filter in Ubuntu and want to reload them. I read the man page and also googled but couldn't find the information. Any help will be appreciated. "This webpage has a redirect loop" in Chrome but working fine on FF and IE? Creating label in QGIS? I am using QGIS 2.18.16 on Windows 10.    What I want is a label to appear within the boundaries of a polygon.    I select a layer, then select command Layer>Labelling, and I get:    From here, I can change "No Labels" to "Show labels for this layer".  But I can't actually create label text.  I find a clue in a prior question:    “Labelling polygon using QGIS?”    The answer was:  The layer you are trying to label must have at least one attribute field populated with data. You can then turn on labelling in Layer Properties and select which attribute field to use for your labels.  I can enter command Layer>Open Attribute Table, and what I get is:    If I need something in the attribute table in order to apply a label, how do I accomplish that?