Continuity of derivative function
Prove that $f'(a)=\lim_{x\rightarrow a}f'(x)$.
Interpreting Granger causality test's results I'm trying to educate myself on Granger Causality. I've read the posts on this site and several good articles online.  I also came across a very helpful tool, the , that allows you to enter your time series and calculate the Granger Stats.  Below, is the output from the sample data included on the site. I have also taken a crack at interpreting the results.   My Questions:     Is my interpretation directionally correct?   What key insights have I overlooked?   Also what is the meaning and interpretation of the CCF charts?  (I'm assuming CCF is cross correlation.)     Here are the results and plots that I have interpreted:  Summary of computational transaction Raw Input   view raw input (R code) Raw Output  view raw output of R engine Computing time  2 seconds R Server    'Herman Ole Andreas Wold' @ wold.wessa.net  Granger Causality Test: Y = f(X) Model   Res.DF  Diff. DF    F   p-value Complete model  356          Reduced model   357 -1  17.9144959720894    2.94360540545316e-05  Granger Causality Test: X = f(Y) Model   Res.DF  Diff. DF    F   p-value Complete model  356          Reduced model   357 -1  0.0929541667364279  0.760632773377753       My interpretation:    Test was based upon 357 data points and was performed with a lag value of 1 The p-value of 0.0000294 means I can reject the null hypothesis that x does not cause y for the Y = f(x). The p-value of .76 allows me to accept the null for X = f(Y) The fact that first hypothesis was rejected and second accepted is a good thing  I'm a little rusty on my F-test so I don't really have anything to say on this for now. I'm also not sure how to interpret the CCF graph.   I really appreciate it if any of you who are well versed with Granger-causality could let me know if I'm interpeting this correctly and also fill in some of the blanks.   Thanks for your help. How do I calculate the date six months from the current date using the datetime Python module? Use character string as function argument Mathematical Induction divisibility $8\mid 3^{2n}-1$ What are the best SEO practices when changing domain names? I'm using the Yoast SEO plugin on a WordPress site on a new, more memorable domain to replace an old static HTML site with no SEO. I've followed , set up 301 redirects from the old domain to the new one, and filed a change of address in the Google Search Console. However, after a few days, the old site still shows in search results, and is ranked higher than the new one if you google our company name.   And, if I try to set the preferred domain in the Google Search Console Site Settings for the new site, I see this in place of the options to prefer www/non-www/http/https:     Not all options are available because of a Change of Address request   related to this site.   I've read of some cases where this condition has lasted for .  Is there anything more I can do to fix this? Have I done something wrong, or do I just need to wait, and for how long? The hidden flaw in "singular they"—what to do about reflexive pronouns? We have  to a question about gender-neutral pronouns, in which he points to the "singular they" and its long history of use in English. (Note that he also advises against using it.) Example:     If someone wants to watch TV tonight, they'll have to do the dishes.   This avoids having to say "he or she" in mixed-gender situations. Okay, fine. I'm not going to get my panties in a bunch if people want to talk this way.  But it occurrs to me that "singular they"—infelicitous at the best of times—really falls apart when extended into the realm of reflexive pronouns:     If someone wants to watch TV tonight, they'll have to do the dishes all by themselves. [?]   That feels very wrong. The only alternative, if one paints oneself into that corner, is to flip it back to singular:     If someone wants to watch TV tonight, they'll have to do the dishes all by themself. [???]   That feels worse.  If I'm to state this as a question, I guess I would put it thus: How can use of a "singular they" truly be reconciled? Is it really as much of a linguistic dead end as it feels to me? Where does gravity get its energy from? I would like to know where gravity gets its energy to attract physical bodies?  I know that the law of conservation states that total energy of an isolated system cannot change. So gravity has to be getting its energy from somewhere, or else things like hydropower plants wouldn't be able to turn the power of the falling water into a spinning rotor.  Just to be clear, Lets create an example:  Lets say we have two objects with equal mass close to each other. So gravity does its job and it pulls each other closer, this gets turned into kinetic energy. This is where I'm lost. According to the law of conservation energy can't be created or destroyed and the kinetic energy comes from the gravitational pull so where does the gravitational pull gets its energy.  If that energy isn't being recycled from some where else then that means you have just created energy, therefore breaking the law of conservation. How should I phrase a question that must be answered with an ordinal number (e.g., the third prime)? I want to make a question having an answer as follows:     5 is the third prime number.   The bold part is the answer. How to phrase the question? Do lens hoods need to be matched to focal length? Is it true that not all generic lens hoods work on all lenses (provided the filter diameter matches, of course)? I read that the focal length plays a role in it - but I cannot find such readings on any lens hoods I see available online. I keep on seeing a fair number of questions (such as ) which get marked as answered within 5-10 minutes of being posted. Sometimes the questions like this are insanely popular and actually see some proper discussion, however often I see questions have few views and only 1 answer. The trouble with these questions is that very few people have scrutinised the answer for correctness, or possibly a better solution. I always make a point of not accepting my answers until a day or so has passed — would it not be a good idea to put a short (1 hour or so) time limit on accepting answers on new questions with few views or votes, or perhaps just an additional confirmation dialog, something along the lines of:  Your question is new and not many people have had a chance to look at it yet.  Are you sure you don't want to wait to see if anyone else has a better suggestion? How do I prove this using question using Cauchy-Schwarz inequality? Why are tar archive formats switching to xz compression to replace bzip2 and what about gzip? Nautilus not opening up, showing GLib error Add warning when changing display name Unwanted behaviour from dict.fromkeys Evolution threads extra lines before signature Prove the map has a fixed point I accidentaly forbid SSH connection to a remote server... What's next? Let's say it again, we all make mistakes, and I have just made one.  A brief history: I was doing some stuff on a VPS (Debian) I'm renting, when I noticed some strange behaviour. Using the netstat command I saw an non-authorized connection through SSH. I didn't know what to do, so I decided to close his connection using iptables:  iptables -A INPUT -p tcp --dport ssh -s IP -j DROP   But I am tired, and I wrote  iptables -A INPUT -p tcp --dport ssh -j DROP   and I kicked myself (and everyone else) out...  How do I fix this? Intuitive idea of the Lipschitz function I'm trying to understand intuitively the notion of Lipschitz function.   I can't understand why bounded function doesn't imply Lipschitz function.   I need a counterexample or an intuitive idea to clarify my notion of Lipschitz function.  I need help  Thanks a lot How do they film continuous blood pooling?