Does there exist a differentiable function $f: \mathbb R \to \mathbb R$ such that its derivative $f'$ is nowhere continuous ?
Could someone give an example of a ‘very’ discontinuous derivative? I myself can only come up with examples where the derivative is discontinuous at only one point. I am assuming the function is real-valued and defined on a bounded interval.
Setting camera clip end via Python A certain physics problem I have been working on has turned into a math problem. Particularly, I want to find the solutions of some equation of the form   $$f(x)=ae^{bx}+cx+d = 0$$  where $a, b, c,$ and $d$ are constant, real numbers that come from the physics problem and $x$ will be a real number. I do not know how to find the solutions of an equation of this form. In the special case where $c=0$, the solution is simply  $$x = \frac{\ln(\frac{-d}{a})}{b}$$  but for other values of $c$, I am stumped. If there is no exact form for the roots, is there a relatively simple way to estimate the roots? How do I install Skype? ContourPlot of implicit function doesn't work when I square the equation How do I calculate the distance between two points specified by latitude and longitude?  For clarification, I'd like the distance in kilometers; the points use the WGS84 system and I'd like to understand the relative accuracies of the approaches available. Suggested improvements to badges index page ()  Currently,  shows the possible badges, number of total recipients, and brief descriptions. However, I have a few suggestions (mostly minor changes) that could make it a bit nicer.   There is no way to see (on that page) which badges I've earned multiple times — they all appear with a green checkmark. It would be nice to show something like "x 4" like in the badges section of the user profile page. For example:      There is no way to see which badges can be awarded multiple times — you have to click on a specific badge to see yea or nay. It would be nice to have a small, simple icon next to badges that can be earned multiple times, similar to the checkmark showing which badges I've earned. The layout would be nicer if the number of recipients for each badge were in a separate column — since the badge images are different sizes, the overlap and ragged right alignment makes the numbers harder to read. An alternative view that groups badges by "achievement level" (gold/silver/bronze) would be nice, as would a view that groups progressive degrees of the same accomplishment. (e.g. Nice Answer &rarr; Good Answer &rarr; Great Answer OR Popular Question &rarr; Notable Question &rarr; Famous Question) Rewiring a switch controlled receptacle to a ceiling light How to give permissions to specific command? I would like to run fastboot command without sudo. it's working if I put sudo in front of it, otherwise it wont.  like this:  $ sudo fastboot devices 2738006a    fastboot  $ fastboot devices no permissions  fastboot   How do I solve this? How to calculate $(G/H)/(K/H)$? I have a group $G=\mathbb{Z}_{36}$ with normal subgroups $K$ and $H$, where $H$ is also a normal subgroup of $K$. What I'm really supposed to find is a bijection between $G/K$ and $(G/H)/(K/H)$ by using the third isomorphism theorem. I have already calculated $G/K$, $G/H$ and $K/H$, but I do not really know how to calculate $(G/H)/(K/H)$. Say that one of the elements in $G/H$ is $\{0,18\}$ and one of the elements in $K/H$ is $\{0,18\}$. Will one of the elements in $(G/H)/(K/H)$ be as follows? $$\{0,18\}+\{0,18\} = \{0+0,18+18\}=\{0,36\}=\{0\}$$ Another way I'm thinking it's possible to caluclate is $$\{0,18\}+\{0,18\} = \{0+0,0+18,18+0,18+18\}=\{0,18,18,36\}=\{0,18\}$$ Is any of these a correct way of calculating an element in $(G/H)/(K/H)$? Does $\pi$ contain all possible number combinations? I came across the :    Which states:     $\pi$ Pi      Pi is an infinite, nonrepeating $($sic$)$ decimal - meaning that   every possible number combination exists somewhere in pi.  Converted   into ASCII text, somewhere in that infinite string of digits is the   name of every person you will ever love, the date, time and manner    of your death, and the answers to all the great questions of    the universe.   Is this true? Does it make any sense ? What is the command to minimize the gnome terminal? How can I minimize the Terminal window using a command? For example: I use the clear command to clear the terminal output, which command can I have to use to minimize the terminal window? proof that translation of a function converges to function in $L^1$ Let $f \in L^1(\mathbb{R})$, for $a\in \mathbb{R}$ let $f_a(x)=f(x-a)$, prove that: $$\lim_{a\rightarrow 0}\|f_a -f \|_1=0$$  I know that there exists $g\in C(\mathbb{R})$ s.t $\|f-g\|_1 \leq \epsilon$, this is also true for $f_a$ and $g_a$.  Now I have the next estimation:  \begin{align*} \|f_a-f\|_1 &amp;= \int_{\mathbb{R}} |f(x)-f(x-a)| dx \\ &amp;= \int |f(x)-g(x)+g(x)-g(x-a)+g(x-a)-f(x-a)| \\ &amp;\leq \|f-g\|_1 + \|f_a-g_a\|_1 + \int |g(x)-g(x-a)| \end{align*} my question is: I can argue that $\lim_{a\to0} \int |g(x)-g(x-a)| = \lim_{a\to0} \lim_{T\to\infty} |g(x_0)-g(x_0-a)| 2T$, now I think that I can change the order of the limits here, but I am not sure why?  P.s $x_0$ is some point in $[-T,T]$, and the above is valid from the intermediate integral theorem for continuous functions, right?  Thanks. How is arccos derived? Stupid question but I need to understand:  If $ \cos = \dfrac{\text{adjacent}}{\text{hypotenuse}} $  What is arcosine? $ \text{adjacent} \cdot \text{hypotenuse}$? Is this the same for arcsine and arctangent? How to loop through all the properties of a class? I have a class.  Public Class Foo     Private _Name As String     Public Property Name() As String         Get             Return _Name         End Get         Set(ByVal value As String)             _Name = value         End Set     End Property      Private _Age As String     Public Property Age() As String         Get             Return _Age         End Get         Set(ByVal value As String)             _Age = value         End Set     End Property      Private _ContactNumber As String     Public Property ContactNumber() As String         Get             Return _ContactNumber         End Get         Set(ByVal value As String)             _ContactNumber = value         End Set     End Property   End Class   I want to loop through the properties of the above class. eg;  Public Sub DisplayAll(ByVal Someobject As Foo)     For Each _Property As something In Someobject.Properties         Console.WriteLine(_Property.Name &amp; "=" &amp; _Property.value)     Next End Sub Prove that $\sup \{-x \mid x \in A\} = -\inf\{x\mid x \in A\}$ I need to prove that $\sup \{-x \mid x \in A\} = -\inf\{x \mid x \in A\}$ and am having trouble moving the $-x$ out of the $\sup$ to $\inf$. Another thing is that I don't quite know how to prove $b = \inf$ (see below). Any help would be appreciated! Here my attempt:  Let $a = \sup\{-x \mid x \in A\}$  Then by definition $a \geq -x$ for $x \in A \implies -a\leq x$ for $x \in A$  Thus $-a$ is a lower bound of $A$. Assume $b$ is a lower bound of $A$  Then $b \leq -a \implies  a \geq -b$ and $a = \sup \{−x \mid x\in A\} = -\inf(A)$ When I search for my address on Google Maps it cannot be found which means I can't order takeaway etc.  Is there a way of adding my address to Google Maps or even add a street address?  I want the address to be publicly available so others can search it. I have a strange issue with Windows 10  When I open .nef files they load fine. However after a second or two the contrast changes noticeably. I tried Windows 10 Photos and Picasa Image Viewer.   What am I trying to accomplish? I just want to open files quickly to check for focus and composition before importing into Lightroom  Lightroom does not have this issue   Here is the example opening with Windows 10 Photos    And here is after a second when it 'fully' loads    Picasa Photo Viewer after it loads a bit. It initially opens fine but then it goes all purple If $X$ is measurable space, $f : X\to [-\infty,+\infty]$, function such that $f^{-1}([-\infty,r))$ measurable for all rational numbers $r$,prove than $f$ is measurable  I know that $f$ is measurable iff set $\{x\in X : f(x)&lt;a\}$ is measurable for all $a\in \mathbb{R}$ but for $a\in\mathbb{Q}$ I don't know how to proceed. I have a small issue with configuring the lidR  for parametrizing the extracted raster files with grid_metrics() function.  My question is how to set in the catalog options that:   be able to overwrite the results with a second run if I want to run my script several times and I do not want to name each output differently. instead of .tif I would like to get .grd files as output.   My current code (which is working) is the following:  normalizedctg = catalog(workingdirectory) opt_output_files(normalizedctg) &lt;- paste(workingdirectory,"/clipped/normalized/{XLEFT}_{YBOTTOM}_height",sep="") height = grid_metrics(normalizedctg, max(Z),res = resolution)   But with a second run on the same code I get the following error:  Error in .getGDALtransient(x, filename = filename, options = options,  :    filename exists; use overwrite=TRUE Upon realizing it is a sarcophagus, he decides to sacrifice himself (I think by disconnecting his air) and exist in it for eternity along with the original creature. It may have been in an old Omni magazine, or some other periodical. I thought it may have been in the anthology "Dead Astronauts" but I did not find the story in there.