A limit with limit zero everywhere must be zero somewhere
Does there exist a function such that $\lim_{x \to a} f(x) = L$ for all $a \in \mathbb R$ but $f(x)$ is never $L$?
How to process all products defined in a project (ConTeXt) If I have  project.tex  \startproject project \product one \product two \endproject   and the corresponding product files, is there an easy way to process all product files, so I get  one.pdf  two.pdf  at once? How do I use variables in a sed command? I tried the following code to replace QQ with ZZ, but it doesn't do what I want:  var1=QQ sed -i 's/$var1/ZZ/g' $file   However, this code does what I want:  sed -i 's/QQ/ZZ/g' $file   How do I use variables in sed? Why are we still using CPUs instead of GPUs? It seems to me that these days lots of calculations are done on the GPU. Obviously graphics are done there, but using CUDA and the like, AI, hashing algorithms (think bitcoins) and others are also done on the GPU. Why can't we just get rid of the CPU and use the GPU on its own? What makes the GPU so much faster than the CPU? Easy to bypass "Are You paying Attention" test by viewing question in detail I think many of us already aware, still I wanted to let you know that First Post and Late Answers review posts are generated on specific time interval. When time comes for post generation I follow some tricks   Click and till a Post appears for review Then right-click and select Open in new tab 5-8 time. Then in every page follow the below steps   Open the Question of post appeared in review in New tab If Page/Question not exists then simply downvote since post was deleted because of -ve response. Else if page exists then   Check the review post's Net Reputation If reputation is -ve then simply downvote the post came in review Else one may select No Action needed.    This way if in any case a post comes for "Are you paying attention" can be handled easily.  I think if a post comes in review should not allow to view post. What are SPF records, and how do I configure them? This is a  about setting up .     I have an office with many computers that share a single external ip (I'm unsure if the address is static or dynamic). Each computer connects to our mail server via IMAP using outlook. Email is sent and received by those computers, and some users send and receive email on their mobile phones as well.  I am using  to generate an SPF record and I'm unsure about some of the fields in the wizard, specifically:   Enter any other domains who may send or relay mail for this domain Enter any IP addresses in CIDR format for netblocks that originate or relay mail for this domain Enter any other hosts which can send or relay mail for this domain How stringent should SPF-aware MTA's treat this?   the first few questions i'm fairly certain about... hope i have given enough info. I have a solid understanding of most OOP theory but the one thing that confuses me a lot is virtual destructors. I thought that the destructor always gets called no matter what and for every object in the chain. When are you meant to make them virtual and why? Change all @username references in comments when a user changes their name? The fact that a user can change their name at will can lead to some confusion in the comments.  For instance, suppose a user named "foo" changes his name to "bar", all those comments signed by "foo" are now signed by "bar", but responses to "bar"'s comments still start with "@foo".  Could those "@foo" references in comments be changed to "@bar"?    A brief discussion on this has set me straight.  I didn't think it through.  Request withdrawn. Lets say we have a dependent variable $Y$ with few categories and set of independent variables.    What are the advantages of multinomial logistic regression over set of binary logistic regressions (i.e. )? By set of binary logistic regression I mean that for each category $y_{i} \in Y$ we build separate binary logistic regression model with target=1 when $Y=y_{i}$ and 0 otherwise. How to make unit selection circles merge? I would like to know how to make this effect of merged circle selection. Here are images to illustrate:     Basically I'm looking for this effect:    How the merge effect of the circles can be achieved ? I didn't found any explanation concerning this effect. I know that to project those texture I can develop a decal system but I don't know how to create the merging effect.  If possible, I'm looking for purely shaders solution. Delete all but the most recent n file for each group of files that share the same prefix in a directory My question is a bit different from some older questions simply asking for "deleting all but the most recent n files in a directory".  I have a directory that contains different 'groups' of files where each group of files share some arbitrary prefix and each group has at least one file. I do not know these prefixes in advance and I do not know how many groups there are.  EDIT: actually, I know something about the file names, that is they all follow the pattern prefix-some_digits-some_digits.tar.bz2. The only thing matters here is the prefix part, and we can assume that within each prefix there is no digit or dash.  I want to do the following in a bash script:   Go through the given directory, identify all existing 'groups', and for each group of files, delete all but the most recent n files of the group only. If there are less than n files for a group, do nothing for that group, i.e. do not delete any file for that group.   What is a robust and safe way of doing the above in bash? Could you please explain the commands step-by-step? Why do we need to use int main and not void main in C++? How can I prove that if $\gcd(a, b) = 1$, then $\gcd(ab, c) = \gcd(a, c) \times \gcd(b, c)$?  By eea there exists $ax+by=1$ from $\gcd(a,b)=1$ so a and be are co-primes there also exists $dk=a$ and $dj= b$ where $d=\gcd(a,b)=1$ this is all the information I have gathered from the question but I dont know how to approach and solve it. Can anyone help explain to me how to arrive at the answer? Thanks! How does Dijkstra's Algorithm and A-Star compare? When transfering data using TLS the browser and server agree the cipher suite to be used - so for example this could be chosen as AES-128 and is (probably) outside of my control. If I separately encrypt a file using AES-256 and then Alice downloads this via her browser (using AES 128) have I still effectively got AES-256 security?  My reasoning goes like this: even if the AES-128 encryption on the TLS link was broken by Eve then all she would get is the AES-256 encrypted file and she would then have to break that encryption to see the plaintext.  Of course the AES-256 key is sent to Alice over a separate channel. Prove $n!>2^n$ for $n\geq4$ using induction. I just want to know if my proof to this question is correct. First, I proved it was true for $n = 4$. $$4!&gt;2^4$$ $$24&gt;16$$ Then, I assumed that it was true for $n=k$. $$k!&gt;2^k$$ Afterwards, I tried to prove it for $n=k+1$. $$(k+1)!&gt;2^{k+1}$$ $$(k+1)k!&gt;(2)(2^k)$$ Since $k!&gt;2^k$, am I allowed to simply do $$(k+1)(2^k)&gt;(2)(2^k)$$ $$(k+1)&gt;2$$ and this is true for all $k\geq3$. Would the last two steps be illegal because I only assumed that $k!&gt;2^k$ is true? I need to do this for displacement mapping, subdivision gives a bad result and makes triangles distinguishable.    Related to:   But I get the error "No other selected object found to use for projection". Ubuntu 12.04 software upgrade does not show 12.10 upgrade Followup to "".  It is argued that even a tiny residual charge would result in huge amounts of electricity in bulk matter, everything would be different, etc. I do not find that a convincing answer: suppose $n$ protons plus $n+1$ electrons are neutral. Why wouldn't we also expect there to be $n$ protons to every $n+1$ electrons? That is, there is no bulk matter problem if every $n$'th atom is a negative ion (for hydrogen).  Now, what empirical lower bound can we give for $n$ in that kind of scenario? Find the Galois group of $x^3-5$ over $\mathbb{Q}$. In this case, the roots of $x^3-5$ are $\{\sqrt[3]{5},\omega\sqrt[3]{5},\omega^2\sqrt[3]{5}\}.$ I think $\mathbb{Q}(\sqrt[3]{5},\omega\sqrt[3]{5})$ is the splitting field of $x^3-5.$ Then, $[\mathbb{Q}(\sqrt[3]{5},\omega\sqrt[3]{5}):\mathbb{Q}]=3,$ I think the automorphisms are: $$id$$ $$\sigma:\sigma(\sqrt[3]{5})=-\sqrt[3]{5}$$ $$\tau:\tau(\omega\sqrt[3]{5})=-\omega\sqrt[3]{5}$$ I'm not sure about what I have done. But how to determine the Galois group based on this? And how to determine the solvability of this polynomial? Reprojecting raster from lat/lon to UTM in R?