Let $f: \mathbb{R} \to \mathbb{R} $ be a continuous function with $f(x+y) = f(x) +f(y),\forall x,y\in \mathbb{R}.$   Find $\frac {df} {dx} $, if it is exist?
Can you show me a continuous function $f \colon \mathbb{R}^n\to\mathbb{R}^m$ that satisfies $f(a+b)=f(a)+f(b)$ but is not linear?   We have that  $$f(0)=f(0+0)=2f(0)\implies f(0)=0\\ f(x-x)=f(0)=f(x)+f(-x)=0\implies f(-x)=-f(x)\\ f(nx)=f(x+x+\dots+x)=f(x)+\dots+f(x)=nf(x)\quad \forall n \in \mathbb{N}$$ But $$ f(-nx)=-f(nx)=-nf(x) $$ So: $$ f(ax)=af(x) \quad \forall a \in \mathbb{Z} $$
What is the significance of the name Bleach? The name "Bleach" seems to be having no relevance to the plot unlike most other series. Was it just chosen at Kubo-sensei's whim or does it have some significance? Maybe some cultural significance associated with shinigami, etc. that I am now aware of? My advisor wants me to include a name of someone who has no contribution to the paper My first advisor wants me to include my second advisor's name in a paper that I worked on before she (2nd advisor) came along.   Should I include her name even though she didn't do anything in that paper? I don't think she even read it. Entering Unicode characters in LaTeX I wonder which of the two is correct:        I have a red and a blue car.   I have a red and a blue cars.      or        Everyone has a left and a right arms.   Everyone has a left and a right arm.      My (non-native) inclination would be for option 1, as it sounds better.  On the other hand, in the following examples, the plural sounds better to me.    Everyone has left and right arms. I have old and new cars. (when I only have only one old car and one new car) The left and the right arms are...   But are they correct? If $A$ and $B$ are compact subsets of $\mathbb R$, then so is $\frac{A}B$. Given compact $A,B \subseteq \mathbb{R}$ with $\alpha := \inf(B) &gt; 0$, define $$\frac{A}B := \left\{\frac{x}y : x \in A  \text{ and } y \in B\right\}.$$ Show that $\frac{A}{B}$ is compact.  I am thinking of using the following result given in a previous exercice but I don't know how to proceed.     $K \subset \mathbb R^n$ is compact if and only if for all $\{x_{n}\}$ such that $\{x_{n} : n\in \{1,2,\dotsc\}\}\subset K$, there exists a convergent subsequence $\{{x_{\phi(n)}}\}$ that converges to a limit $l \in K$.   When does the infimum come into place? How to keep application running after closing the terminal? I have the following annoying problem with Excel 2010. I use Conditional formatting to apply bold formatting to each line of the spreadsheet that meets a condition expressed by a formula. In the "applies to", I write =$A$2:$Q$149, which is the current extent of my spreadsheet. But I want this range to grow with my spreadsheet, i.e., I want the formatting to permanently apply to the whole spreadsheet. Regularly, my "applies to" field is transformed into a complex range, that selects most lines of spreadsheet, but not all, like =$A$2:$Q$138;$A$140:$Q$147. Have you ever encountered such a problem? Is there a solution? I am using a cloned hard drive in attempts to restore some old files.   The cloned drive has 2 users, one with an encrypted home drive, one without. Both are administrators. When I couldn't get into the main user (with the encrypted home directory), I logged in as the 2nd user and changed the password.   It seems that I somehow broke the relationship between the encryption passphrase and the user. I'm pretty sure I have the passphrase. Is there a way to get to the data? Does the diameter of the aperture increase with increases in the focal length with constant F-number(F-stops)? What I have learned from the definition of F-number is that;  The F-number (sometimes also called focal ratio, f-ratio, f-stop, or relative aperture) of a lens is the ratio of its focal length divided by the diameter of the aperture. And F-numbers are expressed as f/1, f/1.4, f/2, f/2.8, f/4, f/5.6, f/8, f/11, f/16, f/22, f/32 etc. Hence F-number (F-stop) depends on two variables; that are diameter of aperture and focal length of the lens.  Aperture diameter/size is one of the component which governs the amount of light that enters to the camera. As far as I known, at different F-stops values (f/1, f/1.4, f/2, f/2.8, f/4, f/5.6, f/8, f/11, f/16, f/22, f/32) the amount of light allowed to enter the camera is fixed and are with either 1/2 or 1/3 stop increment.  This concept is very easy to understand, in case of prime lenses (with a  fixed focal length and variable aperture), where F-stop is inversely proportional to the aperture diameter. For lower F-number (F-stop) the aperture would be bigger allowing more light and for higher Hence F-number (F-stop) the aperture would be smaller allowing less light in to the camera.  F-numbers (F-stops) have been evolved to allow a definite/fixed amount of light in to the camera (As far as I known). In case of zoom lens (with variable aperture and variable focal length), I am not able to understand the what would be the relationship between F-stop and the diameter/area of aperture at different focal length.  I have two related questions  1 . In case of zoom lens,  for example, Canon EF-S 18-135mm f/3.5-5.6  (with variable aperture and variable focal length). At any particular F-number(F-stop), for example at f/5.6 , does the diameter of the aperture increase with increases in the focal length? I mean,  Is the diameter of the aperture at  18mm and f/5.6 would be smaller than that of at 135 mm and f/5.6?   In the above case (i.e. Canon EF-S 18-135mm f/3.5-5.6 ), What is amount of light that would enter the camera at 18mm and f/5.6 compared to that of at 135 mm and f/5.6? Will the amount of lights be same or different? Make Applications (Ubuntu Software) to access all the DRIVES in my computer Inside the alley I found myself cornered by unpleasant sights: walls covered with cracks and peeling paint, storm drains clogged with moss, dumpsters overflowing with bags, cans, and rotten trash. Inside the alley I found myself cornered by unpleasant sights: walls covered with cracks and peeling paint; storm drains clogged with moss; dumpsters overflowing with bags, cans, and rotten trash. Inside the alley I found myself cornered by unpleasant sights. Walls covered with cracks and peeling paint. Storm drains clogged with moss. Dumpsters overflowing with bags, cans, and rotten trash.  Or maybe the three of them are orthographically correct in their respective contexts? EDIT (based on dockeryZ's answer): I think second example is right, regardless of the context. Explanation: . Can Reduce *really* not solve for x here? Example of a finitely generated module with submodules that are not finitely generated I'm looking for an example of a finitely generated module with submodules that are not finitely generated.  I've found  dealing with group (i.e. an example of a finitely generated group with subgroups that are not finitely generated). But I can't figure out whether that question do help to this one.  And I actually want to find a more "module-like" example rather than an example derived from a 'strange' group.  Can you please help? Thank you! Why does light bend towards the normal when passing from a rarer to a denser medium? Whenever light rays are entering a denser medium they bend towards the normal.  Why do rays choose the path towards the normal? Why cannot they choose the path away form the normal? set fontsize for TikZ figure Windows 10 Start Menu can't have multiple shortcuts with the same target Problem  If you create two shortcuts with the same target then only one of them is displayed in the Start Menu. This is true for:   search results within the Start menu manually browsing program folders within the Start menu pinned shortcuts within the Start menu   This is important because:   the shortcuts might have different settings defined in the Properties dialog, such as different compatibility settings, or be set to start in different directories. some programs install a shortcut to a command prompt which is set to open in the program's installation folder. This enables easy access to the program from the command line without having to modify %PATH%, but it means that users who are unaware of this "lose access" to CMD. Programs that do this:           Things I've tried   Renaming shortcuts   Only the shortcut that comes first alphabetically is displayed  Using folders doesn't help   Only the shortcut in the folder that comes first alphabetically is displayed  Pinning shortcuts doesn't help   the pinned shortcut updates to point to the displayed shortcut      Steps to reproduce  Fully up-to-date Windows 10 Home.   Start → search for "Command Prompt" (or "cmd"). (Optional) Right-click on Command Prompt → Pin to Start.   This makes Command Prompt available when you open the Start menu without searching for anything.  Right-click on Command Prompt → Open file location   This opens a File Explorer window in %APPDATA%\Microsoft\Windows\Start Menu\Programs\System Tools  Copy the Command Prompt shortcut and rename the copy "AAA Command Prompt"   This ensures it appears before Command Prompt when files are listed alphabetically.  (Optional) Right-click on the new AAA Command Prompt shortcut → Properties   Change something. (E.g. set the "Start in" directory to a different location, such as C:\)    Notice that it is now not possible to access Command Prompt from the Start menu - it has been replaced with AAA Command Prompt everywhere. If you rename AAA Command Prompt to ZZZ Command Prompt then you get back the ordinary version and lose the customised version. Moving the shortcuts to different folders within %APPDATA%\Microsoft\Windows\Start Menu\Programs\ doesn't help.    Any ideas? Ownership of content in Community Wiki posts Unable to get Excel to recognise date in column I constantly have problems working with dates in Excel, I must be doing something wrong but I don't understand what.  I have a spreadsheet, exported from our exchange server, that contains a column with dates on. They have come out in US format even though I'm in the UK.   The column in question looks like this  04/08/2012 04/09/2009 04/01/2010 04/21/2011 04/05/2012 08/30/2009 08/29/2010 08/28/2011   In Excel, I have highlighted the column and selected Format Cells.... In this dialog box, I have selected the Date, selected English (United States) as the locale and chosen the matching date format from the list. I hit OK and try to sort the data by this column.   In the sort dialogue I choose this column, select sort on Values but the order only gives me options for A to Z, not oldest to newest as I would expect.  This in turn sorts the date data by the first two digits.  I am aware I could re-format this data to ISO and then the A to Z sort would work but I shouldn't have too, I'm obviously missing something. What is it?  EDIT: I messed up the bounty but this should have gone to @r0berts answer, his first suggestion of Text to Columns with no delimiter and choosing 'MDY' as the type of data works. Additionally, if you have a time (i.e. 04/21/2015 18:34:22), you need to first get rid of the time data. However after that the method suggested by @r0berts works fine. User with low reputation shows as having modified thread How can this user , reputation 3, show in  as having "modified 7 hours ago", but on the modified thread, , not show at all?  This user also has two medals, which is somewhat puzzling. As a part of self study, I am trying to prove the following statement:  Suppose $X$ and $Y$ are topological spaces and $f: X \rightarrow Y$ is a map. Then $f$ is continuous if and only if $f(\overline{A})\subseteq \overline{f(A)}$, where $\overline{A}$ denotes the closure of an arbitrary set $A$.   Assuming $f$ is continuous, the result is almost immediate. Perhaps I am missing something obvious, but I have not been able to make progress on the other direction. Could anyone give me a hint which might illuminate the problem for me?