I'm trying to show that a continuous function $f:(a,b) \to \mathbb{R}$ with the mid-point convexity property is convex.  Midpoint convexity is $f((x+y)/2) \leq \frac{1}{2}f(x) + \frac{1}{2} f(y)$. Convexity is $f(tx + (1-t)y) \leq tf(x) + (1-t)f(y)$ for all $t \in [0,1]$.  Starting with $f((x_1 + x_2)/2) \leq \frac{1}{2} f(x_1) + \frac{1}{2} f(x_2)$, I could show   $$f((x_1 + x_2+ x_3 + x_4)/4) \leq \frac{1}{4} f(x_1) + \frac{1}{4} f(x_2)+ \frac{1}{4} f(x_3)+ \frac{1}{4} f(x_4)$$  Using induction I get for all $n$  $$f\left(2^{-n}\sum_{j=1}^{2^n}x_j\right) \leq 2^{-n}\sum_{j=1}^{2^n}f(x_j). $$  I can see that continuity lets me take a limit inside $f()$, but I'm not sure how to continue to prove the general property.
Given that $$f\left(\frac{x+y}{2}\right)\leqslant \frac{f(x)+f(y)}{2}~,$$ how can I show that $f$ is convex. Thanks.      Edit: I'm sorry for all the confusion. $f$ is assumed to be continuous on an interval $(a,b)$.
If I remove Minecraft sound files from .minecraft/assets/objects, will Minecraft be unable to play them? In Minecraft, there are a lot of cool sounds. I decided to find there location and move them to my desktop. I have Ubuntu, and if when I remove them from that folder will Minecraft be unable to play them or is it smart enough to figure out that they are gone and download them again. Before minecraft was installed, it had to download the files. I even moved my favorite song (creative 6) from the sounds. The sounds are not labeled and have seemingly random names like b0b12118a97ddd733a50e9382e91a21652568641 and how will I know what to call them that will be recognized without digging through the index if Minecraft fails to download them? Or do I not have to worry because it will download them again upon launch?  Note: Why does it not label them and call them random names? C# Change A Button's Background Color There is some unwanted extra spaces in between each site in the Your Sites section.  Steps to reproduce:   Touch the Nav Icon in the left top corner Touch the EDIT link in YOUR SITES Scroll the screen; there is unwanted spaces in between the each site   For reference I recorded the same:    Configurations:   App Version: 1.0.85 Device Manufacturer: Motorola Device Model: XT1033 OS Version: 5.1 (2) How do I make sure that a socket bound to a port is properly released on process exit such that the port can be reused without bind() failing with EADDRINUSE? I've written a tiny program which just creates a socket, binds it to a fixed port, waits for a connection and then immediately terminates. When I rerun the program, the bind() call fails with EADDRINUSE, but if I wait a few minutes, it succeeds.  Is there a way I can explicitly "unbind" the socket, thereby freeing the port number? I know a lot of female programmers, and I know there are a good number of them out there.  But I don't recall ever having one of my questions answered by, nor have I ever answered a question by a female programmer here at Stack&nbsp;Overflow.  Now I mostly deal in the Delphi tag, and that may make a difference. But I've got the feeling there is something about the programming social network that Stack&nbsp;Overflow is, along with its form of mini-competition (via reputation) that turns most ladies off.  When I go down the first few pages of top users by rep, there is nary a lady to be found.  Go over to Quantcast and . Go down to the Demographics, and you'll see that a full 20% of the visits are by females. So they are here, lurking and using the wonderful info that Stack&nbsp;Overflow provides.  Then why aren't there more of them participating, both with questions and answers?  What can Stack&nbsp;Overflow do to persuade female programmers to participate more?    So after a dozen downvotes offset by a dozen upvotes, and accusations by many that I am a bigot, the bottom line seems to be that Stack&nbsp;Overflow appeals to males more than females.  This is my observation and I believe a proper analysis would support the hypotheses that a much higher percentage of male programmers use Stack&nbsp;Overflow than the percentage of female programmers.    Assuming that is the case, then Stack&nbsp;Overflow is geared more towards the male psyche than the female. If so, I again ask, what can they do about that?      This question was asked in StackOverflow Meta on November 20, 2009. It was made Community Wiki. It was asked in a constructive manner and should be considered an appropriate question for Meta.   Yet it was closed over two years after it was asked as "not constructive" after more than 40 upvotes and 10 favorites and 16 answers.   Now it resides here on Meta StackExchange, still closed. I just shake my head. I play on the DivineJourney modpack with a friend, but he wants to constantly spawn things in. Is there any way to allow him to be able to use the teleport command, but not be able to spawn in items?  I've already locked the NEI mode to Recipe.  I've looked around for a mod for plug-ins for Forge (as thats what the server is on) but I couldn't find anything thats up to date. I found Sponge, but thats for 1.8+, and I need 1.7.10. How can I count the number of records that have a unique value in a particular field in ROR? Idiom or expression that would imply "someone is a habitual liar" I'm looking for an idiom or expression for describing people who tell lies out of habit (so you cannot trust them at all).   I know that these people are called , but I'm not looking for psychological terms.  Is there any idiom or expression for describing such people?   For example:     "Did X tell that to you?! Come on, don't believe him/her. Don't you know he/she ___ (=is a habitual liar)?" Within my web application directory, how can I change all my folder permissions to rwxr-xr-x (755)? And how can I change all file permissions to rw-r--r-- (644)? I want to change all folders and files including sub-sub files and folders.   I know the following command would change all files and folders but I want to discrimate between a file and a folder.  chmod -R 755 ./my_web_app_project Installing other fonts on wine Show that there are $ a,b \geq 0 $ so that $ |f(x)| \leq ax+b, \forall x \geq 0.$ At what stage should acknowledgements be inserted into the paper if review is not double-blind? When the review process is double-blind, I guess (although I have not read this anywhere) that acknowledgements should not be written in the submitted paper, as this might break anonimity. They should only be inserted into the camera-ready version, after the paper is accepted.  What is the custom when the review process is not double-blind? Should I write the acknowledgements in the submitted version, or wait to the camera-ready version? Why aren't search engines indexing my content? This is a general, community wiki  question and answer pair intended to address any questions   concerning the reasons a site or specific site contents do not appear   in search engine results.      If your question was closed as a duplicate of this question and you   feel that the information provided here does not provide a sufficient   answer, please open a discussion on .   My site (or specific pages on my site) is not appearing in search engine results.  Why isn't my content indexed and what can I do about it? So there are several types of transistors:   BJT JFET MOSFET   Combine all that with the various flavors of each (NPN, PNP, enhancement mode, depletion mode, HEXFET, etc) and you've got a wide array of parts, many of which are capable of accomplishing the same job.  Which type is best suited for which application?  Transistors are used as amplifiers, digital logic switches, variable resistors, power supply switches, path isolation, and the list goes on.  How do I know which type is best suited for which application?  I'm sure there are cases where one is more ideally suited than another.  I admit that there is some amount of subjectivity/overlap here, but I'm certain that there is a general consensus about which category of applications each of the transistor types listed (and those I left off) is best suited for? For example, BJTs are often used for analog transistor amplifiers and MOSFETs are generally used for digital switching.  PS - If this needs to be a Wiki, that's fine if someone would like to convert it for me Is there a word in English to qualify the highest-ranked student in a class in a year?  We are talking here about higher education, especially in the context of curricula that keep a given set of students over the course of several years, being ranked at the end of each year, possibly with competitive admission for the following year. The ranking is a weighted average of all marks obtained over the year.  I am looking for a single word, or expression, to quickly designate the one student with the highest ranking. If there is no such thing in English, then let's try to get the most concise and idiomatic definition.  In all cases, it is not emotionally connoted, i.e. not pejorative nor bragging.  For French speakers: I am looking for the translation of “major de promotion”.    EDIT: additional, not as much of a pure “word request” question: is this anything you should mention on a curriculum vitæ in the English-speaking world? If so, please mention whether the given word is appropriate for such a use&nbsp;&nbsp;:) On Windows 10 Professional, with the November update (Build 10586, Version 1511) , is there any possibility to completely remove Windows Defender ?  I don't want to just disable Windows Defender.  I want to completely uninstall from my system. Best way to save data on the iPhone I am writing an iPhone application, and need to save the state of my application (5K or so).  My main worry is data persisting across upgrades. Some of the applications I use clearly got this wrong, and I would prefer not to! Deserializing JSON when sometimes array and sometimes object Travelling without a visa after passport was stolen Calculating with Mathematica, one can have $$\int_0^{\pi/2}\frac{\sin^3 t}{\sin^3 t+\cos^3 t}\,\mathrm dt=\frac{\pi}{4}.$$   How can I get this formula by hand? Is there any simpler idea than using $u = \sin t$?     Is there a simple way to calculate $$ \int_0^{\pi/2}\frac{\sin^n t}{\sin^n t+\cos^n t}\,\mathrm dt $$ for $n&gt;3$? Could anyone come up with a reference for this exercise?