Let $f(x)$  be a continuous function    on $[a,b]$.       Show that $F(t) = \displaystyle \min_{a\leq x\leq t}f(x)$ is also continuous on $[a,b]$.   Statement seems obvious for me, especially  when I try to draw some special cases of this problem, but I fail to derive a rigorous proof.
I'm working on this problem from Elementary Analysis by Ross which is intuitive when sketched but keeps stymieing me when I try to write it out.  Let $f$ be a continuous function on $[a,b] \subset \mathbb{R}$. Define $f^\star (x)$ as:  $$ f^\star(x) = \sup \{f(y)\mid a \leq y \leq x \} $$ Prove that $f^\star$ is a continuous increasing function on $[a,b]$.  Things I've figured out: (these are mostly trivial)   $f$ is uniformly continuous on $[a,b]$. Since $[a,b]$ is closed and bounded there exists some $c \in [a,b]$ such that $f(c) \geq f(x)\ \forall x \in [a,b]$. (In words: the supremum is actually attained.) If $c \in [a,b]$ is as above, then $f^\star$ is constant (and hence continuous) on $[c,b]$ (which is possibly a singleton).   This seems completely obvious when you actually draw a continuous function but translating that to a formal proof eludes me...
So I came across this exercise: Show that if $Y$ is path-connected then the set of homotopy classes $[I,Y]$ of maps of $I=[0,1]$ into Y has a single point.   But I am confused here. If $Y=S^1$ then not all loops are homotopic! I mean, isn't true that $\pi_1(Y)\subset [I,Y]$? Or maybe I don't understand the set $[I,Y]$ well! How can I upgrade to Java 1.8 on an Amazon Linux Server? I tried sudo yum update but it just keeps java "1.7.0_75". I need 1.8 for it to work with another application but can't figure out how to upgrade it. Do I need to manually install it somehow? There's not much information on this on the internet as far as I can see. Specs:   java version "1.7.0_75" OpenJDK Runtime Environment (amzn-2.5.4.0.53.amzn1-x86_64 u75-b13) OpenJDK 64-Bit Server VM (build 24.75-b04, mixed mode)   When I try update now:     [ec2-________]$ sudo yum update Loaded plugins: priorities, update-motd, upgrade-helper amzn-main/latest                                         | 2.1 kB     00:00 amzn-updates/latest                                      | 2.3 kB     00:00 No packages marked for update   Is there anything else I need to do? Thanks. Proof of the L'Hôpital Rule for $\frac{\infty}{\infty}$ I ask for the proof of the L'Hôpital rule for the indeterminate form $\frac{\infty}{\infty}$ utilizing the rule for the form $\frac{0}{0}$.   Theorem: Let $f,g:(a,b)\to \mathbb{R}$ be two differentiable functions such as that:  $\forall x\in(a,b)\ \ g(x)\neq 0\text{ and }g^{\prime}(x)\neq 0$ and $\lim_{x\to a^+}f(x)=\lim_{x\to a^+}g(x)=+\infty$ If the limit $$\lim_{x\to a^+}\frac{f^{\prime}(x)}{g^{\prime}(x)}$$ exists and is finite, then  $$\lim_{x\to a^+}\frac{f(x)}{g(x)}=\lim_{x\to a^+}\frac{f^{\prime}(x)}{g^{\prime}(x)}$$  My attempt:  Since $\lim_{x\to a^+}f(x)=+\infty$, $$\exists \delta&gt;0:a&lt;x&lt;a+\delta&lt;b\Rightarrow f(x)&gt;0\Rightarrow f(x)\neq 0$$  Let $F,G:(a,a+\delta)\to \mathbb{R}$, $F(x)=\frac{1}{f(x)}$, $G(x)=\frac{1}{g(x)}$. Then by the hypothesis $\lim_{x\to a^+}F(x)=\lim_{x\to a^+}G(x)=0$, $$\forall x\in(a,b)\ \ G(x)\neq 0\text{ and }G^{\prime}(x)=-\frac{1}{g^2(x)}g^{\prime}(x)\neq 0$$ The question is, does the limit $$\lim_{x\to a^+}\frac{F^{\prime}(x)}{G^{\prime}(x)}=\lim_{x\to a^+}\frac{-\frac{1}{f^2(x)}f^{\prime}(x)}{-\frac{1}{g^2(x)}g^{\prime}(X)}=\lim_{x\to a^+}\frac{g^2(x)f^{\prime}(x)}{f^2(x)g^{\prime}(x)}$$ exist?   The limit $$\lim_{x\to a^+}\frac{f^{\prime}(x)}{g^{\prime}(x)}$$ exists by the hypothesis but we don't know if the limit $\displaystyle\lim_{x\to a^+}\frac{g^2(x)}{f^2(x)}$ exists to deduce that the limit $$\lim_{x\to a^+}\frac{F^{\prime}(x)}{G^{\prime}(x)}$$ exists to use the L'Hôpital Rule for the form $\frac{0}{0}$.  EDIT: After discussing it with other users in the site, we came to the conclusion that this proof is only partial and can't logically be continued to yield the Theorem.  As a result, the rule for the $\frac{0}{0}$ form can't be used to proove the rule for the  $\frac{\infty}{\infty}$ form. Mr. Tavares and myself have already given two different proofs (with the pretty much the same main idea) of the Theorem in question using Cauchy's Mean Value Theorem. You can read them below. You can also read the proof Rudin gives for a stronger version of the Theorem (that does not suppose that $\lim_{x\to a^+}f(x)=+\infty$) in his book Principle of Meathematical Analysis. If you have any objections in either proofs please let me know. Thank you. Does a sorcerer's metamagic work for non-sorcerer spells? Can a multiclass sorcerer/cleric use the Metamagic class ability with non-sorcerer spells? For example, using Twinned Spell when casting cure wounds to heal two creatures instead of just one? When I format a drive to NTFS under Windows and plug it into the Ubuntu machine it is perfectly usable, but when I format it under Ubuntu using Disks (delete all partitions and create one single partition with NTFS) it does not show up when plugged into the Windows machine. Please note that in both cases the USB drive contains only one, primary partition formatted to NTFS, for example: /dev/sdb - the actual device /dev/sdb1 - the only, primary partition I've observed this behaviour both with a solid state USB pen-drive as well as an external USB magnetic HDD. How to (re)format a USB HDD or a stick to NTFS under Ubuntu so it is usable under Windows? How do file permissions work? Using math in TikZ Is it true that the light from some galaxies will never reach us?   The explanation for that is that the Universe expanding faster than the speed of light. But, if the speed of light is constant in all reference frames, then in the reference frame of a galaxy, light must move at the speed of light and thus it must reach us (?).   Could you also explain  how is it possible that the Universe is expanding faster than the speed of light if an object can never be observed to move faster than light? (without Doppler effects). What word or phrase best communicates the idea that a human (or other sentient being) is easily swayed, particularly between moral good and evil? Words I have found that are close are:   Impressionable - Easily influenced because of a lack of critical ability.   Implies that the subject lacks critical skills, and misses the concept of good and evil.  Gullible - Easily persuaded to believe something   Misses the concept of good and evil    I created a typical angels and demons scenario to demonstrate the polar extremes of the the spectrum. In this scenario, I lack the word to describe the humans, who can be influenced towards either end of the spectrum if impressed upon by an external entity.     The angels are devoted to good, and are immovable in their ideals. The demons are wicked, and will feign morality only to manipulate others. Both prey upon the humans, who are ____________.   I think that the key here is the spectrum of good an evil. Here are examples of other spectra and their associated words, for comparison.   Easily pushed from calm to angry: Temperamental Easily moved from determined to hopeless: Weak-willed Easily influenced by propaganda: Why does exclamation mark within double quotes cause a Bash error? Please look at these commands:  $ notify-send SYNC TIME! $ notify-send 'SYNC TIME!' $ notify-send "SYNC TIME!" bash: !": event not found $   The first two commands produce a notification bubble as expected. The third gives the error shown.  and  $ echo SYNC TIME! SYNC TIME! $ echo 'SYNC TIME!' SYNC TIME! $ echo "SYNC TIME!" bash: !": event not found $   Here as well, the echo works for first two commands but not in the third.  More problems here (although I was not planning on using this): both notify-send "SYNC!TIME" and echo "SYNC!TIME" give bash: !TIME": event not found.  But both notify-send and echo work with "SYNC! TIME"  Can someone please explain why the bash: !": event not found error appears? How to calculate information included in R's confusion matrix I want know the formulas for the information highlighted in the confusion matrix in the picture below: How many non-negative integer solutions does the equation $3x + y + z = 24$ have? Adding tags to files in OS X: How to use, where are tags stored, and will they work on a network? 8.4 - Use entity browser in a module block from in same way as I would in a custom block Mouse cursor flickering and disappearing I installed Ubuntu today and since the first start up I have a problem with the mouse cursor.  It starts randomly disappearing and flickering a lot.  I have searched for a solution on Google and here but didn't find anything. Does anyone know how to model a depletion mode N channel MOSFET in spice?  I have looked online however the majority of resources have shown how to model a depletion type or enhancement type. I specifically want to model a FET that is fully on when there is a 0 Vgs voltage.  Resources i've looked at How can one prove that $\lim_{n \to \infty}a^{1/n}=1$ for every $a>0$? Prove that $\lim_{n \to \infty} a^{\frac{1}{n}} = 1$ if $a &gt;0$.  In my textbook, we are given a suggestion to let $a^{\frac{1}{n}} = (1+h_n)$ and then show that the $h_n$ term goes to zero using a Theorem that states the following conditions:  $\lim_{n \to \infty} = \infty,$ if $a &gt;1, 1,$ if $a = 1,$ and 0, if $\vert a \vert &lt; 1$. I apologize for not formatting the above cases appropriately; I could not figure out how to use a giant left brace to group them altogether.  If I rewrite it as $a^{n^{-1}}$, then I thought that this might help, but I don't think the Binomial Theorem would then apply since the exponent is negative. Furthermore, the problem is only concerned with the positive, real $n$-th roots. I am quite stuck on this problem and any suggestions or advice would be greatly appreciated.   I am using the textbook Introduction to Analysis by Arthur Mattuck. How to make a half cylinder? I am having a tough time building a half cylinder and have not found any tutorial showing me how to do that. Live USB for Ubuntu 13.10 does't recognize Windows 8 OS Is there a single word that describes both Witches and Wizards? In the Harry Potter world, there's the term "muggle" for people with no magical abilities, "muggle-born" for those with magical abilities who's parents are muggles and "squib" for those with no magical abilities who's parents are magical.  But is there any collective term for all magical people (as opposed to wizards for males and witches for females)?  I don't recall seeing any word that refers to all men and women and children with magical abilities, and it's the only group left without noun or collective noun to describe them.