Most of the functions that I have seen have their discontinuities on rationals and continuities on irrationals!  I am wondering if there is any exampe of some function whose continuities are rationals? Or is other words      The set of continuities of a function $f:\mathbb{R}\to\mathbb{R}$ can be $\mathbb{Q}$?
I have a question about subsets $$ A \subseteq \mathbb R $$ for which there exists a function $$f : \mathbb R \to \mathbb R$$ such that the set of continuity points of $f$ is $A$. Can I characterize this kind of sets? In a topological,measurable or in some way? For example, does there exist a function continuous on $\mathbb Q$ and discontinuous on the irrationals?
Increasing the maximum number of TCP/IP connections in Linux How do I figure out who's been banished to the penalty box recently? Egg replacer for bread dough? I've been reading (most) questions and answers (here) about baking bread (which I love) and how to replace eggs in many recipes (which I wish I didn't need). Turns out, I too have egg allergy (not too severe) and a few of my favorite breads require one or more eggs.  So far I've seen two options, in many threads, that look promising: Ener-G egg replacer and replacing eggs with flax. However, not a single use case I've seen here is about bread. I'm looking for an option to replace 1-3 eggs in a bread dough with: 500 g of white wheat flour, 200 ml of milk, 50-80 g of butter, a variable amount of sugar (5-50 g) and 20-30 g of fresh yeast.  Examples of this dough are the , the  and a similar recipe from Galicia (northwest Spain) with even more eggs. And so many other breads, but these are my favorite ones :)  Has anyone tried flax or ener-g on a similar bread dough? Electron shells in atoms: What causes them to exist as they do? I have seen similar posts, but I haven't seen what seems to be a clear and direct answer.  Why do only a certain number of electrons occupy each shell? Why are the shells arranged in certain distances from the nucleus? Why don't electrons just collapse into the nucleus or fly away?  It seems there are lots of equations and theories that describe HOW electrons behave (pauli exclusion principle), predictions about WHERE they may be located (Schrödinger equation, uncertainty principle), etc. But hard to find the WHY and/or causality behind these descriptive properties.  What is it about the nucleus and the electrons that causes them to attract/repel in the form of these shells at regular intervals and numbers of electrons per shell?  Please be patient with me, new to this forum and just an amateur fan of physics. Why (in relatively non-technical terms) are Calabi-Yau manifolds favored for compactified dimensions in string theory? How does ownership in fractional shares work? I've been trying to find out some more information about fractional shares, and how they work. I understand conceptually, you own x% of a share rather than x "full" shares, but I'm curious about how such a system is implemented.  As I understand it, in a "traditional" purchase of shares, the role of a broker is to act as an executor on your behalf, and does not hold any assets themselves. In the case of a fractional share, is this then not the case? For example, the broker holds the complete shares, and then just ensures that they keep an internal record of each partial owner in their system? So there would be no external record of partial ownership?  Thanks! How do I attach rope to a mesh I am trying to make a pirate ship and am trying to attach rope to the sails. my rope is made using a path, and soft body physics. I can not find a way to attach my rope to the sail. and I can not use pinned vertices because this is not a cloth modifier. I'm tutoring high school students. I've always taught them that:     A charged particle moving without acceleration produces an electric as well as a magnetic field.   It produces an electric field because it's a charge particle. But when it is at rest, it doesn't produce a magnetic field. All of a sudden when it starts moving, it starts producing a magnetic field. Why? What happens to it when it starts moving? What makes it produce a magnetic field when it starts moving? Executing SSIS 2012 package that has script components from external application Pointwise convergent and total variation I recall Asimov had a character who was The autoc of , clearly based on the word “autokrat” but I don’t recall the exact spelling.  This stumped Google, which would rather correct the spelling to some other word than find a unique word.  In the story, he had gone undercover and the reader did not know either that some other character was in fact the Autoc, until it had been revealed to the main character. How do you hide back faces so that you only see faces when its normal is facing the camera? When exporting it for use in a game, the face is rendered in only the direction of its normal, so this would be useful to have something similar. Let $x$ be a real number such that $x &gt; 0$. Prove that x +$\frac {1} {4x}  ≥ 1$.   Not really sure on the correct way to approach it/is valid and could use some help.  Answer:  Proof Strategy: Proof by cases:   $x = 1$ $x &gt; 1$ $x &lt; 1$   --  Case 1: $x = 1$  $x+\frac1{4x}$  $= (1) + 1/(4(1))$  $= 1.25 \ge 1$  Case is true  Case 2: $x &gt; 1$, in this case $x = 2$.  $x+\frac1{4x} $  $= (2) + 1/(4(2))$  $= 2.125 \ge 1$  Case is true  Case 3: $x &lt; 1$, in this case $x = 0.5$.  $= (0.5) + 1/(4(0.5))$  $= 1  \ge 1$  Case is true  Since all possible cases were satisfied therefore $x+\frac1{4x} \ge 1 $ when $x &gt; 0$. Probability that uniform random variables equal the natural numbers I have random variables $X_{1}, X_{2}, \dots X_{n}$ that are each Unif$\{1, \dots, n\}$, and I want to calculate $$P(\{X_{1}, X_{2}, \dots X_{n}\}=\mathbb{N})$$  My intuition is that this means each $X_{n} = n$, which occurs with probability $\frac{1}{n!}$.  Does this seem correct? Flag as Broken Link I keep finding broken links in posts, currently I add a comment in the hope someone will return to update it or if I find a link I edit the answer.   I imagine this will only increase over time so I wonder if there could be a standard way to manage this?  My idea is adding a new flagging reason like "contains broken links", then it will go to a review queue where people will be able to have a look and fix the broken links. Every contractible space X is simply connected because X is homotopy equivalent to a point.  Is there a direct proof of this fact? There obviously is a (free) homotopy between any loop and the trivial loop at the base point. But how to construct a based homotopy, which is required for a loop to be trivial in the fundamental group? I'm curious to know how important the bias node is for the effectiveness of modern neural networks. I can easily understand that it can be important in a shallow network with only a few input variables. However, modern neural nets such as in deep learning often have a large number of input variables to decide whether a certain neuron is triggered. Would simply removing them from, e.g., LeNet5 or ImageNet have any real impact at all? In all the discussions about how the heavy elements in the universe are forged in the guts of stars and especially during a star's death, I usually hear that once the star begins fusing lighter atoms to produce iron (Fe) that's the end of the star's life and the whole system collapses onto itself; and based on how massive the star was initially, it has different outcome - like a white dwarf, a neutron star or a black hole. I have rarely heard a detailed explanation of how the elements heavier than iron are produced. I would appreciate a convincing explanation of this process. My number is associated with iMessage on other phone, how to remove it? Good day everyone. For only 1 hour (!) I used my relatives iPhone with my sim card and that's it, he keeps receiving all of my iMessages. I tried checking in iTunes all devices associated with my account and his device is not listed. I tried to log in and log out to/from iMessage on his phone with his apple ID, still won't help. How can I stop this nightmare!? When I go to iMessage settings>Send &amp; Receive on his device, there is only my phone number listed, how do I remove it without smashing phone to the wall?! Thanks and have a great day  EDIT. After talking to apple support for some time, they removed my phone number from my friend's iMessage. So I guess if you are facing same problem you have to contact support. How to create a soft or symbolic link?