An increasing function $f:B\to\mathbb{R},\ B\subset\mathbb{R}$ must be continuous at every element of $B$ except for a countable subset of $B$ I am trying to prove the following statement, but I have been stuck for a while and I am looking for an hint about how to prove it: &quot;Suppose $B\subset\mathbb{R}$ and $f:B\to\mathbb{R}$ is an increasing function. Prove that $f$ is continuous at every element of $B$ except for a countable subset of $B$.&quot; What I have tried to do: Since I wasn't able to tackle the original statement I tried by assuming that $B$ is also a Borel set so as to exploit in some way the fact that $f$ must then be a Borel measurable function. Then if $D:=\{x\in B: f\text{ is not continuous at }x\}$ and $C$ is a Borel set we have that $f^{-1}(C)$ is a Borel set and $f^{-1}(C)=(f^{-1}(C)\cap D)\cup (f^{-1}(C)\cap B\setminus D)$ and at this point I tried to go on by contradiction, assuming that $D$ is uncountable and tried to show that $(f^{-1}(C)\cap D)\cup (f^{-1}(C)\cap B\setminus D)$ is not a Borel set but I haven't succeeded so I would greatly appreciate any hint or comment or explanation that could nudge me towards a more fruitful approach, thanks.  EDIT (Proof): Let $D:=\{x\in B:f\text{ is not continuous at }x\}$: since $f$ is monotone (increasing)  so if $d\in D$ then $I_d:=(\lim\limits_{x \to d^-,\ x\in B\\}f(x),\lim\limits_{x \to d^+,\ x\in B}f(x))=(\sup_{x&lt;d,\ x\in B}f(x),\inf_{x&gt;d,\ x\in B} f(x))$ is a non-empty interval in $\mathbb{R}$ hence there must be a rational number $q_d$ in it which we can choose and since $I_d\cap I_{d'}=\emptyset$ for $d\neq d'$* this rational number is also unique so we can define an injective function $g:D\to\mathbb{Q}, g(d):=q_d$. Thus $\#(D)\leq \#(\mathbb{Q})=\#(\mathbb{N})$ ie the set of points at which $f$ is discontinuous is (at most) countable, as desired. $\square$  (*) Suppose $I_d\cap I_{d'}\neq\emptyset$ for some $d\neq d'$ (we can suppose wlog that $d&lt;d'$): then there must be $y\in I_d\cap I_{d'}$ so $\sup_{x&lt;d,\ x\in B}f(x)&lt;y&lt;\inf_{x&gt;d,\ x\in B}f(x)$ and $\sup_{x&lt;d',\ x\in B}f(x)&lt;y&lt;\inf_{x&gt;d',\ x\in B}f(x)$ which implies (since $d\leq d'\Rightarrow \inf_{x&gt;d,\ x\in B}f(x)\leq \sup_{x&lt;d', x\in B}f(x)$) that $y&lt;y$, contradiction.
How to show that a set of discontinuous points of an increasing function is at most countable I would like to prove the following:       Let $g$ be a monotone increasing function on $[0,1]$. Then the set of points where $g$ is not continuous is at most countable.     My attempt:    Let $g(x^-)~,g(x^+)$ denote the left and right hand limits of $g$ respectively. Let $A$ be the set of points where $g$ is not continuous. Then for any $x\in A$, there is a rational, say, $f(x)$ such that $g(x^-)\lt f(x)\lt g(x^+)$. For $x_1\lt x_2$, we have that $g(x_1^+)\leq g(x_2^-)$. Thus $f(x_1)\neq f(x_2)$ if $x_1\neq x_2$. This shows an injection between $A$ and a subset of the rationals. Since the rationals are countable, $A$ is countable, being a subset of a countable set.    Is my work okay? Are there better/cleaner ways of approaching it?
Number of homomorphisms between two cyclic groups. Isomorphism of a product $C_n \times C_m$ of cyclic groups with the cyclic group $C_{mn}$ Given that $C_n$ is a cyclic group of order $n$, what conditions must integers $n$ and $m$ satisfy such that the group $C_n \times C_m$ is isomorphic to C$_{mn}$?   So I attempted to investigate a few lower-order cyclic groups to find a pattern. In particular, I looked at $C_2 \times C_3$ and compared it to $C_6$. Here are a couple of diagrams:  $C_2 \times C_3$ -   $C_6$ -   They Cayley tables look pretty different (identity elements are all over the place for $C_2 \times C_3$) so I concluded that they are not isomorphic.  I did the same for $C_2 \times C_4$ and $C_8$:  $C_2 \times C_4$ -   $C_8$ -   Again, I arrived at the conclusion that the 2 groups are not isomorphic due to their Cayley table structure.  It seems I'm stuck now. Are my conclusions wrong for the examples I examined? If so, how do I find the conditions for the general case of comparing $C_m \times C_n$ with $C_{mn}$? I have a hierarchical category tree and checkboxes next to each item that allow to enable/disable the correpsonding category.    I assume it is an expected default behavior that all subategories of natural environment become checked when I check the natural environment category. But what about unchecking it again: Should this automatically uncheck all subcategories or restore the previous state in case some were checked already?  Furthermore... Should unchecking one of the subcategories cause the parent category to become unchecked? Cleaning my Macbook Pro Retina leaves greasy 'streaks' across the screen Syncing folders outside Dropbox using Symbolic Links and Junctions I've been reading a lot of information on this and yet I'm still unsure if its possible.  What I want to achieve is: sync a folder that is outside Dropbox and any changed to that folder or the files within get mirrored to the Dropbox folder and the Dropbox server.  This is what I've tried: I've install Link Shell Extension. Picked a folder with 10 pictures inside and created a junction for it inside the Dropbox folder. When I delete a pictures from inside the original folder (outside Dropbox) the change is reflected in the Junction folder inside Dropbox Folder, not on the server.  The exact same behaviour with the Symbolic Link option.  If I manually select Pause Syncing from Dropbox system tray icon and then Resume syncing = it uploads the changes to the server.  I know I can move all my files inside Dropbox folder and create the junctions/symlinks at their original locations but I want to do it the other way around.  Also what is the difference between choosing the Junction and Symbolic Link Option in regards to a folder?  Thank you. Writing function definition in header files in C++ I have a class which has many small functions. By small functions, I mean functions that doesn't do any processing but just return a literal value. Something like:  string Foo::method() const{     return "A"; }   I have created a header file "Foo.h" and source file "Foo.cpp". But since the function is very small, I am thinking about putting it in the header file itself. I have the following questions:   Is there any performance or other issues if I put these function definition in header file? I will have many functions like this. My understanding is when the compilation is done, compiler will expand the header file and place it where it is included. Is that correct? Is there a way to execute JavaScript in Perl? Book where wizards learn magic by bonding with crystals that imbue them with powers I read this book a long time ago, but I'm guessing it was written sometime in either the 80's or early 90's. It was a rather long/large book.  Plot summary: Set in a fantasy land of wizards and magic, some kids (maybe) are recruited to learn magic. In the process they find out that wizards are created by being placed in a crashed spaceship that bonds them with crystals that imbue the powers. One of them is able to bond with many/multiple crystals to become very powerful. My question was closed for being a "shopping list question", even though it was very much related to the topic of the site. Why? Pass by Reference / Value in C++ I would like to clarify the differences between by value and by reference.  I drew a picture     So, for passing by value,  a copy of an identical object is created with a different reference, and the local variable is assigned the new reference, so to point to the new copy  How to understand the words: " If the function modifies that value, the modifications appear also within the scope of the calling function for both passing by value and by reference "  Thanks! When testing the difference between 2 proportions why do we use a z-test rather than a t-test?    Further, is there a simple way to conduct an omnibus test for significant differences between more than 2 proportions (in the form of percentages). Is there an equivalent to a one-way ANOVA for this?  I imagine that you could use logistic regression (assuming you have the original data for the proportions in the form of 0s and 1s) Are there other options? Asking the position of a person in a sequence Jordan normal form for a characteristic polynomial $(x-a)^5$ Write down all the possible Jordan normal forms for matrices with characteristic polynomial $(x-a)^5$. In each case, calculate the minimal polynomial and the geometric multiplicity of the eigenvalue $a$. Verify that this information determines the Jordan normal form.  I found this question in a textbook that I'm using for a test I have tomorrow. I think that I need to use the method for finding the Jordan normal form of a matrix but I can't see how to apply it and I don't have much intuition about the answer...  I'm guessing that there are 5 possibilities since the minimal polynomial can be any factor of the characteristic, but I don't know how to prove this.  Some help would be great for my test tomorrow! Thanks Can we deploy PROFILES alone from one sandbox to another? What should I do if I want to be able to run a given program regardless of my current directory? Should I create a symbolic link to the program in the /bin folder? How to block specific notifications from an app? A certain app I use gives me crucial notifications that I need, but also notifies me almost daily that over 80% of my RAM is used. I would like to allow this app to send me all notifications except those that contain the string "80% of your RAM" in the text of the notification.  Obviously, there is no way to filter a certain app's notifications in stock Android (other than blocking all of a certain app's notifications, but that's not what I want!), but is there a way that I can block notifications if they contain a certain string of text?  Nexus 6P running Android Marshmallow 6.0.1, if you're curious the app in question is . Homotopy equivalence of universal cover As part of am exam question (Q21F ), I'm trying to prove that if $X$ and $Y$ are path-connected, locally path-connected spaces with universal covers $\widetilde{X}$ and $\widetilde{Y}$, respectively, then if $X \simeq Y$ then $\widetilde{X} \simeq \widetilde{Y}$.  My attempt might be correct, but it seems too complicated and amongst the jumble I may have made some incorrect assumptions. So what I'm looking for is (a) verification or correction; and (b) ideas as to how I could simplify my argument.    My argument:  Let $p:\widetilde{X} \to X$ and $q:\widetilde{Y} \to Y$ be the covering maps, and let $X \overset{f}{\underset{g}{\leftrightarrows}} Y$ be a homotopy equivalence.  Fix a point $\widetilde{x} \in \widetilde{X}$, let $x=p(x) \in X$ and $y=f(x) \in Y$, and fix $\widetilde{y} \in q^{-1}(\{y\}) \subseteq \widetilde{Y}$. Also let $x' = g(y) \in X$ and fix $\widetilde{x'} \in p^{-1}(\{x'\}) \subseteq \widetilde{X}$.  Define a map $\widetilde{f} : \widetilde{X} \to \widetilde{Y}$ as follows. For $\widetilde{z} \in \widetilde{X}$ let $\widetilde{u}:[0,1] \to \widetilde{X}$ be a path from $\widetilde{x}$ to $\widetilde{z}$. Let $z=p(z)$ so that $u=p\widetilde{u}$ is a path in $X$ from $x$ to $z$. Let $v=fu$, so that $v$ is a path in $Y$ from $y$ to $f(z)$. Lift $v$ to a path $\widetilde{v}$ in $\widetilde{Y}$ with $\widetilde{v}(0) = \widetilde{y}$. Define $\widetilde{f}(\widetilde{z}) = \widetilde{v}(1)$.  Notice that $q\widetilde{v}=v$ so that $q\widetilde{f} = fp$.  Define $\widetilde{g}:Y \to X$ analogously: For $\widetilde{z} \in \widetilde{Y}$ let $\widetilde{a}:[0,1] \to \widetilde{Y}$ be a path from $\widetilde{y}$ to $\widetilde{z}$. Let $z=q(z)$ so that $a=q\widetilde{a}$ is a path in $Y$ from $y$ to $z$. Let $b=fa$, so that $b$ is a path in $X$ from $x'$ to $g(z)$. Lift $b$ to a path $\widetilde{b}$ in $\widetilde{X}$ with $\widetilde{b}(0) = \widetilde{x'}$. Define $\widetilde{g}(\widetilde{z}) = \widetilde{b}(1)$.  Likewise, notice that $p\widetilde{g}=gq$.  Claim: $\widetilde{X} \overset{\widetilde{f}}{\underset{\widetilde{g}}{\leftrightarrows}} \widetilde{Y}$ is a homotopy equivalence.  We have $p\widetilde{g}\widetilde{f} = gq\widetilde{f}=gfp \simeq p$ and $q\widetilde{f}\widetilde{g} = fp\widetilde{g} = fgq \simeq q$.  This shows that $\widetilde{g}\widetilde{f}$ and $\widetilde{f}\widetilde{g}$ are covering translations (deck transformations) and are therefore homotopic to the respective identity maps. So we have a homotopy equivalence.    Any comments would be appreciated. I did some mathematical induction problems on divisibility   $9^n$ $-$ $2^n$ is divisible by 7. $4^n$ $-$ $1$ is divisible by 3. $9^n$ $-$ $4^n$ is divisible by 5.   Can these be generalized as $a^n$ $-$ $b^n$$ = (a-b)N$, where N is an integer? But why is $a^n$ $-$ $b^n$$ = (a-b)N$ ?  I also see that $6^n$ $- 5n + 4$ is divisible by $5$ which is $6-5+4$ and $7^n$$+3n + 8$ is divisible by $9$ which is $7+3+8=18=9\cdot2$.  Are they just a coincidence or is there a theory behind?  Is it about modular arithmetic? Adjusting space between array rows and columns How to calculate $\lim_{n\to\infty}(1+1/n^2)(1+2/n^2)\cdots(1+n/n^2)$? How can we compute the following limit:     $$\lim_{n\to\infty}(1+1/n^2)(1+2/n^2)\cdots(1+n/n^2)$$   Mathematica gives the answer $\sqrt{e}$. However, I do not know to do it.