Suppose $f'(x)&gt;0$ in (a,b). Prove that $f$ is strictly increasing in $(a,b)$, an let $g$ be its inverse function. Prove that $g$ is differentiable, and that $g'(f(x))=\frac{1}{f'(x)}\text{ (a&lt;x&lt;b)}$.  I've managed to prove the second part, but have difficulty proving that $g$ is differentiable. We know that for some neighborhood of $t$, we have $$\mid\frac{f(x)-f(t)}{x-t}-f'(t)\mid&lt;\epsilon$$ I want to bound $$\mid\frac{g(f(x))-g(f(t))}{f(x)-f(t)}-\frac{1}{f'(t)}\mid$$ by using the above inequality. I only know that $g(f(x))=x$ and $g(f(t))=t$. But then I'm not sure how to proceed. Could you tell me if this is the right approach, and give me any hint to manipulate this inequality? Thanks!
So, I was given the following problem as part of a homework assignment.     Suppose $f'(x) &gt; 0$ in $(a,b)$. Prove that $f$ is strictly increasing in $(a,b)$, and let $g$ be its inverse function. Prove that $g$ is differentiable, and that   $$g'(f(x)) = \frac{1}{f'(x)}$$   I have proven that $f$ is strictly increasing in $(a,b)$, and I could prove that $g'(f(x)) = 1/f'(x)$ if I could prove that $g$ is differentiable. The problem is that I am having trouble with a proof of that. Any advice?   Also, as a reference, this is exercise 5.2 from Baby Rudin.
Rendering fire animations with transparent background I'm creating a simple fireball sprite for my game and I want to render a fire animation.  I would like to get the upper images with transparent background, but when is set the transparency on under Render -> Film one of my frames will look like lower pictures.  You can see the difference of these two images. How do I redirect in a class sub-classing the EventSubscriber class? To tell about a guess about a case in the past, why use "would+have+past participle" instead of "must+have+past participle? I have seen this sentence in an article on bbc, which is about how people lived thousands of years ago. &quot;When this Neolithic home was built, it would have had a cosy roof of turf..&quot; When I read it, I got confused. Because the fact seems as a simple past situation or guessing/assumption about something in the past. it simply gives information about a past event, eg &quot;when I was a child, I had a blue toy car. I have been making a lot of research on whether or not this structure &quot;would+have+past participle&quot; could be used instead of the simple past or &quot;must+have+past participle&quot;, but I could not find any satisfying explanation. Almost all grammer contents cover this structure is used when we imagine the opposite of what actually happened in the past eg. &quot;referring to a missed opportunity&quot; (eg. &quot;if you had come to the party, you would have seen Tom.&quot;), But, in this case of this sentence, &quot;type 3 conditional&quot; is irrelevant. It is not an if sentence,  It simply gives information about how this house was like or was probably like when it was built long ago. Just like it would be If and old lady would want to tell about &quot;how she was like when she was a child years ago. she would simply say &quot;when I was a child, I had long curly hair.**. So &quot;when I was a child, I would have had long curly hair.&quot; would be wrong. So, my question is: 1-Does this structure &quot;would+have+past participle&quot; completely replace simple past sentences? or can be it be used instead of simple past? 2- If this structure was simply used because the author was not %100 sure about the fact as it was a long long time ago -so he wanted to make a guess or assumption, why did he not simply use &quot;must+have+past participle&quot; or &quot;it was likely that blah blah&quot; which must be used for guessing or assumptions about past happenings. Would it not have made more sense if it was like: &quot;When this Neolithic home was built, it must have had a cosy roof of turf. "He didn't know where New Jersey was" As you may know, Facebook allows users to choose a "vanity URL," allowing easy linking to one's profile. However, once set, this URL cannot be changed and this was done for security reasons, understandably.  I have a friend whose now ex-partner set the URL to be "herfirstname.hislastname" without her permission. She tried to change it after this, but to no avail.  Has anyone successfully contacted Facebook and got them to either disable the name and revert back to an ID or change the name to another? Cardinality of the set of all real functions of real variable How does one compute the cardinality of the set of functions $f:\mathbb{R} \to \mathbb{R}$ (not necessarily continuous)? How do I find out which program and process ID accesses a given IP address in Windows? Is it possible to catch, which program is accessing a specific IP address?   I found my computer is flooding and wish to check addresses one by one.  May be it is possible to set up some audit for this? Mouseover tooltip for delete count is wrong I would like to know what would be the best way to do unit testing of a servlet.   Testing internal methods is not a problem as long as they don't refer to the servlet context, but what about testing the doGet/doPost methods as well as the internal method that refer to the context or make use of session parameters?  Is there a way to do this simply using classical tools such as JUnit, or preferrably TestNG? Did I need to embed a tomcat server or something like that? If $(a,b)=1$ then prove $(a+b, ab)=1$. Show that every connected graph with $n$ vertices has at least $n − 1$ edges.  How can I prove this?  Conceptually, I understand that the following graph has 3 vertices, and two edges:  a-----b-----c  with $a$, $b$ and $c$ being vertices, and $\{a,b\}$, $\{b,c\}$ being edges.  Is there some way to prove this logically?  --UPDATE--  Does this look correct?  Any advice on how to improve this proof would be appreciated.  Thank you. Change array key without changing order How to see python symbolic names of objects on cursor hover of menu items in Blender 2.8 in Blender 2.79 one could hover over a UI element and see the symbolic python name of said element. How would one accomplish the same with Blender >= 2.8? Effective marketing strategies for independent game projects I would really like to hear how other independents generally approach marketing their game with limited resources. I know how critical marketing is since I've seen it first hand on several fronts. I've been part of teams that didn't put an ounce of marketing into their project and didn't come close to meeting expectations. I've also poured a lot of time into visiting forums to discuss our previous title which resulted in minor bumps over time.  I really think  is doing some amazing grass roots marketing with the way they are managing their community around their Blog. They've massed a following that is willing to pay (far in advance) them for their game before they finish it to help support them. That's awesome.  But how many people are really willing to divulge such information that early? Should that be a concern? Should I start talking about my next project before I really have anything implemented?   What has worked for you? I would be particularly interested to hear how people approach Apple's iTunes store. I believe we're looking at around 33,000 games on the store these days. I want to do a search on StackOverflow for a few terms, but also by user, since I know which user referenced the item I'm looking for. Ubuntu16.04 boots into initramfs How to understand the Ubuntu file system layout? In Windows there’re perhaps only a couple of important folders (by important I mean important in my logical picture of the Windows file system) in the installation drive (in my case C:\). Namely Program Files and Windows. I simply stay away from Windows folder and the “add remove program files” is good enough to handle the program files folder of Windows. Of course there’s a folder named Users where the users (who are not admins) can access only their folders.  Thus there’s a clear picture at some level in my mind of the Windows file system. In Ubuntu, when I reach the location /, there’s a huge list of folders, most of which I have no clue as to what they contain. The /bin folder seems to be the equivalent of the Windows folder in windows. The /usr folder seems like it’s the equivalent of the Users folder in Windows. But even the /home folder looks like it can fit the bill.  Please understand that I do understand, that Ubuntu (Linux) has a different character than that of Windows, i.e., there need not be exact equivalent of Windows functions, in Ubuntu. All I am looking for is a bit more clearer picture of the Ubuntu file system.   This question is a part of a bigger question which I am splitting up to make it more answerable. The original question can be found here: David studies linguistics during the flight to LV-223. We get to listen and read a fable. Is it written in a real language? What is the name of the language? Can the fable be translated?  Approximate quotation:     hjewis jasma hwaelna nahast akwunsez dad'kta, tam ghermha vagam ugenthe JSP in /WEB-INF returns "HTTP Status 404 The requested resource is not available" How to use custom icons with mapKit framework?