We got $$ f(x) = \begin{cases} \sin \frac1x,&amp;x\neq 0 \\ 0,&amp;x = 0 \end{cases} $$  I have to prove that g is continuous, where $ g: \mathbb R \rightarrow \mathbb R, x \rightarrow xf(x)$. How can I do this?
Given the function: $$f(x) = \begin{cases} x\cdot\sin(\frac{1}{x}) &amp; \text{if $x\ne0$} \\ 0 &amp; \text{if $x=0$} \end{cases}$$  Question 1: Is $f(x)$ continuous at $x=0$?  Question 2: What is the derivative of $f(x)$ at $x=0$ and how do I calculate it?
Let $q_1,q_2,q_3,q_4$ be orthonormal vectors in $\mathbb{R}^4$,$z_1,z_2,...,z_6$ orthonormal vectors in $\mathbb{R}^6$ and $A=z_1q_1^T + z_2q_2^T$. Let $q_1,q_2,q_3,q_4$ be orthonormal vectors in $\mathbb{R}^4$,$z_1,z_2,...,z_6$ orthonormal vectors in $\mathbb{R}^6$ and $A=z_1q_1^T + z_2q_2^T$.  a) Find the base and dimension of fundamental subspaces of matrix A  b) Using the results given by a) find the general solution for $Ax - z_1$.  Any hint would be appreciated. Format a date in XML via XSLT How does the Golden Ratio/Spiral Work? Is it useful with Logo Design? Google Chrome is my favorite browser, and I can't imagine going on without it. Suddenly (without closing and re-opening) the browser; chrome has been reading characters upside down, very weird:    Chrome version:    I tried restarting my browser and my MAC (running OS X Yosemite 10.10), but the error keeps on coming. extracting energy from cosmological expansion This question is a more concrete reincarnation of an .  Are there mechanisms to extract energy from the cosmic rate of expansion? putting some extremely long ropes across the universe and extract work from their increase of separation?  assuming this is possible ( i don't see why it would not ), what form would take the back-reaction force? in all physics, every time you extract work from some system by some force, this system will exert a force back. Can we slow down cosmic expansion by variations of these mechanisms? 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? I am carrying out research to look at differences in fish density and fish species richness when using two different underwater visual census methods. My data was originally count data but then typically this is changed to fish density but I have still decided to use a Poisson GLM, which I hope is right.   model1 &lt;- glm(g_den ~ method + site + depth, poisson)   My 3 predictor variables are method, site and depth which I ordered as factors when I input them.   My response variables are grouper species richness, grouper density and the same for other fish groups. I am aware that density is not an integer and it is numerical data e.g 1.34849. I am now however getting this error:      In dpois(y, mu, log = TRUE) : non-integer x = 0.037500   I've been reading up and many people suggest using an offset, is this the most advisable thing to do? How can we get the command line of a running application? I came across this formula in a list:  The number of ways of distributing $n$ identical objects among $r$ groups such that each group can have $0$ or more $(\le n)$ objects  I know that standard way of doing this is to solve the problem of distributing n identical objects and $(r-1)$ partitions among themselves which can be done in $C(n+r-1,r-1)$ ways.   But I am unable to prove to myself why it is not $(r+1)^n$.  Because each of the n objects has r+1 choices, either group1, group2,... group r or none at all. I just saw now that a new build had rolled out in which the comment upvote and flag options are always visible. Previously, one had to mouse over the area in order to see them.  Why was this just changed? Is it to support touch screens? I'll admit, it was difficult to find the flag button using my touchscreen laptop running Microsoft Edge.  Cross-site duplicate on MSO: How to display a video in a web page using html,php how to get user permissions for nfs share? How can I create that glitter tears effect and the sparkles? Order of a product of subgroups. Prove that $o(HK) = \frac{o(H)o(K)}{o(H \cap K)}$. I am having Problems converting user rights. I want to assign the following rights to certain files/folders but dont know hat to translate them into a format which chmod will work with like chmod 755.  -rwxrwxrwx    will be?  And  drwxr-xr-x   will be?  And finally  -rw-r--r--   will be? Delayed choice entanglement swapping: why are Alice & Bob's measurements useless without Victor's? Formatting doubles for output in C# Solve this integral for free WiFi I saw this today, I checked in Mathematica and the integral comes out to $\pi$, but I have no idea how to solve it.     FREE Wi-Fi: The Wi-Fi password is the first $10$ digits of the answer.   $$\int_{-2}^2\left(x^3\cos\frac x2+\frac12\right)\sqrt{4-x^2}\ dx$$ How to *disable* automatic reboots in Windows 10? Windows 10 lets you 'schedule' a reboot for later. I want to disable it.  Evidently Windows scheduled itself for a reboot last night when I wasn't looking and just closed everything I had been working on the night before.  I reboot on the regular; I don't need Windows to do that for me.  Can I disable it completely? I don't mind if it downloads everything, and then says "hey, you should reboot," but it should never reboot itself, ever.  I'm using the "Pro" edition of Windows 10. What is the reason for the top secret two space newline markdown weirdness?