I plan to use the fact that if $f$ is monotone and non constant  $[a,b]$, then it is continuous on $[a,b]$ if its range $R_f=\{f(x)|x \in [a,b]\}$ is the closed interval with endpoints $f(a)$ and $f(b)$. I dont know how to go from here.
A function $f: \mathbb{R} \to \mathbb{R}$ is said to have the intermediate-value property if for any $a$, $b$ and $\lambda \in [f(a),f(b)]$ there is $x \in [a,b]$ such that $f(x)=\lambda$.  A function $f$ is injective if $f(x)=f(y) \Rightarrow x=y$.  Now it is the case that every injective function with the intermediate-value property is continuous. I can prove this using the following steps:   An injective function with the intermediate-value property must be monotonic. A monotonic function possesses left- and right-handed limits at each point. For a function with the intermediate-value property the left- and right-handed limits at $x$, if they exist, equal $f(x)$.   I am not really happy with this proof. Particularly I don't like having to invoke the intermediate-value property twice.  Can there be a shorter or more elegant proof?
Show $\lim_{h\to 0} \frac{(a^h-1)}{h}$ exists without l'Hôpital or even referencing $e$ or natural log The current is maximum through those segments of a circuit that offer the least resistance. But how do electrons know beforehand that which path will resist their drift the least? How do I get an animated gif to work in WPF? What control type should I use - Image, MediaElement, etc.? How to change default user in WSL Ubuntu bash on Windows 10 I often end up issuing complex commands over ssh; these commands involve piping to awk or perl one-lines, and as a result contain single quotes and $'s.  I have neither been able to figure out a hard and fast rule to do the quoting properly, nor found a good reference for it.  For instance, consider the following:  # what I'd run locally: CMD='pgrep -fl java | grep -i datanode | awk '{print $1}' # this works with ssh $host "$CMD": CMD='pgrep -fl java | grep -i datanode | awk '"'"'{print $1}'"'"   (Note the extra quotes in the awk statement.)  But how do I get this to work with, e.g. ssh $host "sudo su user -c '$CMD'"?  Is there a general recipe for managing quotes in such scenarios?.. Password Reset for Users on a Multisite Subsite What is the smallest (installed size) ubuntu based distribution? For ridiculously small hard drives (8GB), what Ubuntu Based (although not necessarily official) distribution offers the smallest installed size?  Let me clarify: I still want a certain amount of personal file space, preferably a few gig. I also don't have a problem with performance, while it is mediocre, a 2.6GHz single-core and 1GB of ram suffice to run ubuntu fine, but i find that it  uses 6-7GB of space. I am aiming for closer to 4GB. How do I run specific sudo commands without a password? On one particular machine I often need to run sudo commands every now and then. I am fine with entering password on sudo in most of the cases.  However there are three sudo commands I want to run without entering password:   sudo reboot sudo shutdown -r now sudo shutdown -P now   How can I exclude these commands from password protection to sudo? Compare two dates with JavaScript Label rows of a matrix by characters How to add label to index the rows and columns of the matrix ? because I want to number the line outside the matrix       \[     \begin{array}{lc}       \ Mat_{\varPhi to M} &amp; \left(\begin{array}{@{}ccccc@{}}             1 &amp; 1 &amp; 1 &amp; 1 &amp; 1 \\             0 &amp; 1 &amp; 0 &amp; 0 &amp; 1 \\             0 &amp; 0 &amp; 1 &amp; 0 &amp; 1 \\             0 &amp; 0 &amp; 0 &amp; 1 &amp; 1 \\             0 &amp; 0 &amp; 0 &amp; 0 &amp; 1               \end{array}\right) \\[15pt]     \end{array}     \] No hyper refs from citation to bibliography using BiBTeX, hyperref and harvard C++ Struct Pointer Segfault I'm trying to find a way of building a new power supply for my workshop, in order to keep it small and not so hot I will be using a spare xformer to step down mains to 24v and then a variable buck converter to adjust it (and CC mode, displays and blinky stuff).   I thought it would be a good idea to use a linear regulator on the output stage to smooth things out, but I can't seem to think of a control scheme that allows me to change the voltage using only one potentiometer. The buck stage is this topology but with a external PNP transistor.(It has already been prototyped, it works alright, I know the mc34063A is old)   The only idea I got that ""works"" is using the buck converter output as a reference for a linear regulator (IC or discrete parts), but using a SMPS output as reference for anything is a bad idea, due to the amount of noise and fluctuations.   This is the sort of output stage I can and have built (or bought) over time. Just a big discrete regulator.  In short: How to control a MC34063A step down (buck) converter so it follows the output stage +1-3 V of headroom. What do I need to install LED strip lighting? I've been looking to install LED strips, either RGB or white for lighting and ambiance. It wouldn't be the main lighting of the room, so no necessity for high power LEDs.  The perimeter of the room is a little less than 25 meters. Looking around in online stores, I saw that the reels come in maximum lengths of 5 meters, so I would need five of those.  What I'm not sure is how to connect the strips together.  I was looking at , or  with .  If I choose the white LEDs, what power supply, controller, and connector would you recommend to be able to daisy chain the strips?  If I choose the RGB LEDs, what kind of data repeaters would work to be able to control all the strips at once? I am looking for system monitoring tools which are GUI and CLI or web-based which include basic functions such as:   CPU Usage Ram Usage Swap Usage Disk Usage ( Space / I/O ) Heat Monitoring   I know there are many tools I can use, but I am looking for a single tool that has these basic functions. How can I get admin access to a Mac without knowing the current password? Generate three correlated uniformly-distributed random variables SF novel -- the only human on a space station full of aliens Some things I remember:   Pretty sure the author was female The novel had won some award or other The title was one word (I think) and it started with "I" (I think) It would have come out in the mid to late 90s or early 2000s It was a paperback or trade paperback, in English.  The book was originally written in English as far as I know. The cover, if I remember correctly, had an orangish background.   The plot involved a sole human representative to what was effectively an alien "United Nations" type of organization.  There were myriad types of aliens, many of whom were completely biologically incompatible.  Even the ones that could live in the same conditions generally found each other either repugnant or incomprehensible.  The main character ended up feeling incredibly lonely and was "set up" with a superficially similar humanoid female, but found their pheromones were incompatible and it was still difficult to communicate. Problem: Prove that $1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$ for $n \in \mathbb{N}$.  My work: So I think I have to do a proof by induction and I just wanted some help editing my proof.  My attempt:  Let $P(n)=1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$ for $n \in \mathbb{N}$.  Then $$P(1)=1^2=\frac{1(1+1)(2+1)}{6}$$ $$1=\frac{6}{6}.$$ So $P(1)$ is true.  Next suppose that $P(k)=1^2+2^2+\cdots+k^2=\frac{k(k+1)(2k+1)}{6}$ for $k \in \mathbb{N}$. Then adding $(k+1)^2$ to both sides of $P(k)$ we obtain the following: $$1^2+2^2+\cdots+k^2+(k+1)^2=\frac{k(k+1)(2k+1)}{6}+(k+1)^2$$ $$=\frac{2k^3+3k^2+k+6(k^2+2k+1)}{6}$$ $$=\frac{2k^3+9k^2+13k+6}{6}$$ $$=\frac{(k^2+3k+2)(2k+3)}{6}$$ $$=\frac{(k+1)(k+2)(2k+3)}{6}$$ $$=\frac{(k+1)((k+1)+1)(2(k+1)+1)}{6}$$ $$=P(k+1).$$ Thus $P(k)$ is true for $k \in \mathbb{N}$. Hence by mathematical induction, $1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$ is true for $n \in \mathbb{N}$. Inverse of function is continuous or not?