Let $\Omega \subset \Bbb R^n$ be a convex set and $f: \Omega \to \Bbb R$ be continuous. Show that $f$ is convex if and only if for all $x,y \in \Omega$, $$f\left (\frac{1}{2}x + \frac{1}{2}y \right ) \leq \frac{1}{2} f(x) + \frac{1}{2} f(y)$$ Give a counterexample to the same statement without the assumption that $f$ is continuous. $\Rightarrow$ is obvious but I am not getting any clue for the other side. I was trying to proceed using the bisection method e.g. for $f(\lambda x+(1-\lambda )y)$ I was trying to approach $\lambda f( x)$ by halving the numbers say $\frac{\lambda x+(1-\lambda )y}{2}$ and same with $(1-\lambda)f( y)$ but it seemed to be not working. Please help me with the proof and the counterexample.
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 $(a,b)$.
Why is a suspicious program draining my battery with GPS usage? iOS 11-13 - unable to paste into comments I’m able to copy-paste from any application to any application. I’m able to copy from a comment or from a question or from an answer. I’m able to paste to a question or answer. However, if I attempt to paste into a comment, nothing happens. I am able to subsequently paste into another application, confirming that the text is actually on the clipboard.   However, copying SE links will paste them successfully, but all other text - including other links - still fails. This leads me to believe that it’s a SE issue and not an Apple one.   This has been an issue since 11.0 beta and persists through iOS 13 release. How to kill the process using the name of the program instead of PID? I started my Python program in the background using nohup as mentioned below -   nohup zook.py &amp;   Now I am trying to kill this process so I did the ps command as mentioned below  root@phxdbx1145:/home/david/zook# ps ax | grep zook.py 16352 pts/6    S+     0:00 grep --color=auto zook.py   But somehow, everytime its PID getting changed, I don't know why. Whenever I do like this -   kill -9 16352   It always say, No Such Process.  And when I do px command again, I see that PID got changed automatically..  So I am not sure how do I kill this process?  Is there any way I can kill the process with the name somehow?  UPDATE:-  This is what I am getting. I did pkill -9 zook.py and then I did ps command as mentioned below and it is shwoing zook.py constantly?  root@dbx1145:/home/david/zook# pkill -9 zook.py  root@dbx1145:/home/david/zook# ps ax | grep zook.py 23870 pts/6    S+     0:00 grep --color=auto zook.py  root@dbx1145:/home/david/zook# ps ax | grep zook.py 23872 pts/6    S+     0:00 grep --color=auto zook.py  root@dbx1145:/home/david/zook# ps ax | grep zook.py 23874 pts/6    S+     0:00 grep --color=auto zook.py  root@dbx1145:/home/david/zook# ps ax | grep zook.py 23876 pts/6    S+     0:00 grep --color=auto zook.py Would it be possible to create a wand more powerful than the Elder Wand? How do you find a question's view counter? How do you see how many views a question has had when you're not in the newest category?  I've tried a word-for-word search for the question, but still can't see the view counter. Is string in array? What would be the best way to look in a string[] to see if it contains a element. This was my first shot at it. But perhaps there is something that I am overlooking. The array size will be no larger than 200 elements.  bool isStringInArray(string[] strArray, string key) {     for (int i = 0; i &lt;= strArray.Length - 1; i++)         if (strArray[i] == key)             return true;     return false; } Remove particularly tricky items from context menu I've found various sources of info about removing these annoying things which I never use... but I haven't succeeded in getting rid of the following.    (Avira-related:) - Scan selected files with Avira  (I believe all this is M$ crapware... I presume it's all been lovingly crafted so they're all impossible to remove. You never know, though: some super-uber-uber-geek might just have found a way) - Restore previous versions  - Cut - Copy - Delete  - Rename Pass An Instantiated System.Type as a Type Parameter for a Generic Class Using Google Music Manager in Linux from the command line My goal is to use Google Music Manager on a Linux server.  I'm wondering if the music manager can operate from the command line and with no GUI.  If I could set the software up to watch a particular directory on the server and upload from there, that would be the only necessary functionality.  Thank you. Is it OK to call a professor by his first name when he/she signs emails by only first name? If a professor in a North American country presents his- or herself by his/her first name in email messages, does this mean that students can refer to him/her by his/her first name? Or is this generally not a good idea, unless the professor has explicitly mentioned that he/she can be referred to by first name? I've noticed that most professors, who prefer to be referred to more formally, do not sign their emails with just their first names, but usually initials or first and last name. When I was a kid we had a VHS tape of some Disney live action (not animated) sci-fi movie. It may have been a mini-series or a made-for-tv movie, but I'm not sure. I'm also not sure about the year but it would probably have been late 80s or early 90s based on when I would have seen it. I know it was Disney because they had some live action Goofy and Mickey running around the movie's set as an introduction discussing the plot with a host before the movie.  The plot is basically as follows:   A starship is loaded with mostly children and teenagers as a cross-galaxy human settling effort. Accidents (which are not really accidents and later revealed) cause adult crew members to be killed.  They travel to another planet and face all sorts of troubles. It arrives at the new planet after overcoming obstacles.   Other memorable scenes that may help:   Navigating out of earth's asteroid belt made out of space junk. One of the adults was killed via an airlock malfunction. The ship was supposed to be peaceful with no weapons. At one point they bring on board some crazy guy who puts together a rail gun. There's a memorable line like "Do you want no weapons at all or a well aimed rock?"  There's a planet or space station (I can't remember which) that has a fight scene with the evil bad guy. This scene takes place on what looks to be a carnival or fairground. The bad guys are trying to steal the ship the kids are on because it has some kind of really good futuristic space drive.   Hopefully you remember it better than I do! I've tried looking online for it before with no luck. How do I extract a single chunk of bytes from within a file? LLC-SCorp to self - actually, does the corp pay payroll taxes? In the USA. Here are the two common scenarios for the self-employed:  A. Joe makes 100,000 over the year. It's paid to him as 1099 payments from his various clients.  Joe pays the many taxes on that.  B. Joe makes 100,000 over the year.  He has an LLC as SCorp (or maybe an SCorp per se).  The clients pay $100,000 to the entity JoeCorp.  JoeCorp pays Joe a certain salary (it might be the whole 100,000, or, it might be (let's say) half of that and the rest is taken as profit distribution, not relevant here).  So, JoeCorp (his LLC) pays Joe say $50,000.  In fact: does JoeCorp have to pay "payroll taxes" on that??  Does it perhaps make a difference if it is an LLC with S-Corp election v. S-Corp?  (I realize there are many questions on here about "1099 v. corp" .... but I've never really understood this and can't seem to google it up.) Inequality $(1+\frac1k)^k \leq 3$ How can I elegantly show that: $(1 + \frac{1}{k})^k \leq 3$ For instance I could use the fact that this is an increasing function and then take $\lim_{ k\to \infty}$ and say that it equals $e$ and therefore is always less than $3$   Is this sufficient? What is a better wording than "increasing function" Mannequin Duplication glitch How to automatically generate a stacktrace when my program crashes Android Seekbar with two thumbs Are both of the following sentences correct?  a: You can call me, if you need me.  OR  b: You can call if you need me.  Note that in a:, the comma is placed before the "if" and is not present in case b.  From this link, I gather that it isn't necessary because it's a short sentence:   But can someone point out an "official" source on this usage? I've made a roughness 0 glass pane, when I'm in lookdev mode it's reflecting a background that I didn't put there. I know I could just ignore it, but I'd rather be able to see what it'll mostly look like when I'm in the later stages. I also just want to know how on Earth I managed to set this background without realising it  Reflection completely ignores any objects I put in front of it. If I placed this glass inside my house, it would still just reflect trees (This is the point where I felt I couldn't ignore it - reflections are going to play a big part in my project and I don't want to be in render mode all the time)  The nodes are the default glass BSDF, except I set roughness to 0  The glass is set to 20mm thick, if that makes a difference  The background moves with my perspective and moves when I relocate the glass  I have a few HDRI's on my PC, but none look remotely like this   Also, I'm using cycles as I find Evee buggy as hell Windows 8.1 missing font files after restart Some days ago I rebooted my windows 8.1 system. After the reboot all my system texts were very cryptic - showing only some strange symbols. After some investigation I discovered that my c:\windows\fonts folder was nearly empty. So I copied the font files from another windows 8.1 installation to the nearly empty folder. As soon as I had copied the files the system texts were readable again.  But after another reboot the problem occured again - the systems font folder was empty again. Does anyone know why windows deletes my font files during a restart?