Let $f: \mathbb R \rightarrow \mathbb R$ defined by $$f(x) := \begin{cases}x^2 \sin \frac 1 x\ &amp; x \neq 0\\ 0\  &amp; x = 0\end{cases}$$  Show $f$ is differentiable on $\mathbb R$:  Let $\epsilon &gt; 0$.  $x_0 \neq 0$: $\left\lvert\frac {x^2 \sin \frac 1 x - x_0^2 \sin \frac 1 {x_0}}{x-x_0} \right\rvert \le \epsilon$   I'm thinking that since the fraction is continuous there exists $\delta &gt; 0$ satisfying the above for $x \in [-\delta,\delta] / \{0\}$ ?  $x_0 = 0$: $\left\rvert\frac {x^2 \sin \frac 1 x - 0} {x} \right\rvert \le \epsilon$   How can I evaluate this limit ? I cannot say the fraction is continuous at $0$ ?  Also I must find the derivative of $f$ and show it is not continuous at $x=0$, but I'm stuck.
Let $f(x) = x^2 \sin{\frac{1}{x}}$ for $x\neq 0$ and $f(0) =0$. (a) Use the basic properties of the derivative, and the Chain Rule to show that $f$ is differentiable at each $a\neq 0$ and calculate $f'(a)$. You may use without proof that $\sin$ is differentiable and that $\sin' =\cos$.  Not even sure what this is asking.  (b) Show that $f$ is differentiable at $0$ and that $f'(0) =0$.  $\frac {f(x)-f(0)}{x-0} \to \lim_{x \to 0} x \sin(1/x)$. $x \sin(1/x) \leq |x|$ and $\lim_{x \to 0} |x|=0$. Thus $f(x)$ is differentiable at $0$; moreover $f^{'}(0)=0$.  (c) Show that $f'$ is not continuous at $0$.  $f{'}(x)=x^{2} \cos(1/x) (-x^{-2}) + 2x \sin (1/x)$. In pieces: $\lim_{x \to 0} \cos (1/x)$. $f^{'}(0-)$ nor $f{'}(0+)$ exists as $x \to 0$ $f^{'}(x)$ oscillates infinity between $-1$ and $1$ with ever increase frequency as $x \rightarrow 0$ for any $p&gt;0$ $[-p,0]$, $[-p,p]$ or $[0,p]$ $f$ is not continuous. Question: How to show more rigorously?
How to solder with one hand? I am a beginner at electronics.  I don't have many tools and stuff, so I am forced to solder by hand.   I don't have a pcb to solder on, so I usually just solder the wires together, but it's really difficult without a holder.  Is there an easier way to do this? If not, is there a substitute for a wire holder?  Any help would be appreciated. Any way to change my review of suggested edit? I goofed and rejected a suggested edit, a vote that was actually directed at the previous edit that I was reviewing. As this is unfair to the user who posted the suggested edit in question, is there any way in which to change/cancel my reject? How to call a confirmation dialog box? I have a button in my menu named "New File" as shown:    and I want when I click on the "New File" button to imitate what the Blender File->New does, where a check message box appear first confirming that you want to reload the Start-up File and if pressed then the startup file is loaded as shown below.    This is my script, when I press on new it opens the startup file without checking:  class NewFile(bpy.types.Operator):     """New File"""     bl_idname = "wm.newfile"       bl_label = "New File"     bl_options = {'REGISTER', 'UNDO'}      def execute(self, context):         bpy.ops.wm.read_homefile()         return {"FINISHED"} Probability that 2 appears at an earlier position than any other even number in a permutation of 1-20 Suppose we uniformly and randomly select a permutation from the 20! Permutations of 1,2,3,...,20. What is the probability that 2 appears at an earlier position than any other even number in the selected permutation?  My approach :   2 in first position ==> 1*19! (1 is the position of 2)  2 in second position ==> 10 *1 *18!  2 in third position ==> 10*9 *1 *17!  . . .  2 in 11th position ==> 10*9*8*7*6*5*4*3*2*1*1*9!  But this was asked as a multiple choice question in GATE 2007 and the options were  (A) 1/2   (B) 1/10  (C) 9!/20!  (D) None of these  I am not able to reduce this analysis to a given option. How to check if installing a package will require a reboot? I often install packages and updates for them, although sometimes it is not convenient to do a reboot if one of the packages needs it, so is there any way of telling BEFORE installing the package/update whether or not it will require you to do a reboot? I have already looked  and the solutions there don't solve my problem, nor the more or less same one described there. Let $n \geq 1$ be an odd integer. Show that $D_{2n}\cong \mathbb{Z}_2 \times D_n$. Let $n \geq 1$ be an odd integer. Show that $D_{2n}\cong \mathbb{Z}_2 \times D_n$. I define a map $$\phi:D_{2n} \rightarrow \mathbb{Z}_2 \times D_n$$ by $R \mapsto (0,r^{\frac{n+1}{2}})$ and $M \mapsto(1,m)$. Then I am stuck at showing the map is bijective. By the way, do we need to show that $(0,r^{\frac{n+1}{2}})(1,m)(0,r^{\frac{n+1}{2}})^{-1}=(1,m)$ ? If we can prove this, then what can we conclude? Passing multidimensional arrays as function arguments in C In C can I pass a multidimensional array to a function as a single argument when I don't know what the dimensions of the array are going to be?  Besides, my multidimensional array may contain types other than strings. I need a step-by-step, simple and easy way to configure swappiness. I think it would be super if, in my LaTeX editor, I could click on the name of package...  \usepackage{blob}   ... and the corresponding style file, blob.sty, would be opened.  Do any LaTeX editors have this feature? I mainly use Aquamacs/AucTeX myself, but am interested in other editors too. I reckon it would be mildly challenging to implement, because blob.sty might be    in the current working directory,  in the system texmf directory, or  in the user's texmf directory. Find a recurrence relation for the number of $n$-digit binary sequences with no pair of consecutive $1$s.  I know my base case:$$a_1 = 2$$ for either $1$ or $0$. Normally to construct the recurrent relation I would say fill in $n-1$ spaces then add a $1$ or $0$, which is simply:$$a_n = 2\cdot a_{n-1}$$However, I don't believe this is right, because this doesn't take into account that there can't be any consecutive $1$s.  Even when I use the base case of $2$ digits:$$a_2 = 3$$for either $01$, $10$, or $00$, I still get the recurrence relation of:$$a_n = 3\cdot a_{n-2}$$which also does not account for $01$ and $10$ occurring consecutively.   So how would I express this? Given the Python function:  def a_method(arg1, arg2):     pass   How can I extract the number and names of the arguments. I.e., given that I have a reference to func, I want the func.[something] to return ("arg1", "arg2").  The usage scenario for this is that I have a decorator, and I wish to use the method arguments in the same order that they appear for the actual function as a key. I.e., how would the decorator look that printed "a,b" when I call a_method("a", "b")? A guide to moderating comments Where can we find frequently updated details about travel/entry restrictions due to Coronavirus (nCoV-19)? Defining the Product of Ideals Where does the &quot;naive definition&quot; of the product of two ideals $I$ and $J$, $IJ = \{ ij \mid i \in I, j \in J \}$ fall apart? (The product of two ideals $I$ and $J$ is defined to be $IJ := \sum_i a_ib_i$, where each $a_i$ is in $I$ and each $b_i$ is in $J$.)  Note: This is a question from a problem set I'm working on. I am not expecting a full solution for the answers, but I'd like some help knowing where to look. It seems to work fine in $\mathbb{Z}$ and $\mathbb{C}[x]$. Exam with 18pt main font How to retrieve a module's path? If $a,b$ are positive integers such that $\gcd(a,b)=1$, then show that $\gcd(a+b, a-b)=1$ or $2$. and $\gcd(a^2+b^2, a^2-b^2)=1$ or $2 $ What are good strategies for booster drafts? I know that doing a booster draft can be somewhat complicated, because to really be good you need to know a lot about the set you're drafting (or be really good at evaluating cards quickly) and to pay attention to signal cards.    Besides going into the whole complicated strategies of knowing the set and paying attention to signal cards, what are some relatively simple/basic guidelines for drafting? SQL-Like Inner Join on IDs in Two Layers in QGIS Using QGIS v2.6.0-Brighton.  This question is not about editing layers geographically in anyway. It is merely trying to find data from two layers as you would in an SQL Query and append an extra ID to the second layer.  I have one polyline layer each identified with two different IDs. There is a point layer also with one of these IDs, I wish to find this corresponding ID and append them as a column to the point layer.  How do I do this without finding each row individually and typing it in manually (i.e. larger scale)? I would prefer to use SQL like syntax but if there is a GUI as well that would be ideal. Happy to install official plugins. Open Windows Terminal as admin with WIN+R