Is it possible to say that the directional derivatives of a function f at a exists but f is not differentiable at a? If so, why? I cannot get the intuition about it. Could someone please elaborate on this point a little bit?  I am self studying mostly, so I need to discuss these trivial matters with someone :)  Thanks in advance!
Consider the function : $$f: \mathbb{R}^2 \rightarrow \mathbb{R} , (x,y) \mapsto   \begin{cases}         0 &amp; \text{for } (x,y)=(0,0) \\    \frac{x^3}{x^2+y^2}       &amp; \text{for } (x,y) \neq (0,0)   \end{cases} $$  Show that $f$ not differentiable at $(0,0)$ but all directional derivatives exist.  I don't know how to tackle this problem. Can someone give some hints or solve the problem? Thanks :)
Weird artifacts in my UV mapped texture Any ideas what's causing those chunks of geometric artifacts?  Argh. Smallest dense subset of $\mathbb{R}$ I am not sure if what I am looking for even makes sense (or) exists. Anyway I would be happy if someone can clear my confusion.  The set of real numbers $\mathbb{R}$ is obtained as completion of $\mathbb{Q}$. However, $\mathbb{Q}$ is not the only set which is dense in $\mathbb{R}$. $\mathbb{Q} \backslash \mathbb{Z}$ is also a dense subset of $\mathbb{R}$. I am wondering if it makes sense to talk of the "smallest" dense subset of $\mathbb{R}$. To phrase what I am looking for precisely and what I mean by smallest, I am looking for a dense set $A$ of $\mathbb{R}$ such that if $B$ is a proper subset of $A$ then $B$ is not dense in $\mathbb{R}$. I am able to "see" that the set $A$, I am looking for doesn't exist since any open interval consists of infinite rationals. But I am unable to precisely argue out to myself and convince why $A$ doesn't exist. Could some one throw more light on this? Sum over all non-evil numbers A flying fly inside a sealed box on a scale suppose there is a scale able to measure weight with an uncertainty of $10^{-9}kg$ . On the scale, an airtight plastic chamber is placed. Initially, a fly of mass $10^{-5}kg$ is sitting at the bottom of the chamber, which sits on the scale. At a later point in time the fly is flying around the chamber. Will there be a difference in the observed weight as measured by the scale when the fly is sitting at the bottom of the chamber compared to when it is flying around the chamber at some point in time? If so, what does the value of this difference depend on (I am most concerned with the case where the fly has not touched any surface of the container in enough time for the scale to reach some equilibrium value (or do the pressure variations induced from the flies wings cause constant fluctuations in the scale)? Planetary/Atmospheric volumetric clouds I created a planet in blender a while ago, and now i'm interested in adding volumetric clouds instead of using a 2D texture. I considered adding a bump node to my material, but then it would remove the realism in the image.  This is what I've created:    This is the effect I want to create:      Any thoughts? Prove that if $g^2=e$ for all $g$ in $G$ then $G$ is Abelian. This hurts my eyes to read it and my ears to say it, but the writer stands by          About the same number of people was awarded bachelor's degrees in 2010 as filed for personal bankruptcy (1.6 million).   Is the subject in this sentence singular or plural? Wifi hotspot software package other than ap-hotspot? I have tried using ap-hotspot to create a wifi-hotspot but it gives me some problem every time I use it . I just want to ask is there any other software  package that create a wifi-hotspot othen than ap-hotspot ? "He is loved", is 'loved' an adjective or a verb? He is loved.   This is something that I've always kind of wondered.  In a sentence like this, is loved a verb or an adjective? Can it be considered either? Force Messages to use SMS for specific contact rather than iMessage? Essentially, I would like to set my phone so that all my outgoing messages to a specific friend, with whom I currently text back and forth over iMessage, are sent out over SMS, never iMessage. Additionally, I would like to block any incoming iMessage texts of his and (hopefully) they'll fail and be resent via SMS.  Let me restate that: I would like to force SMS for both parties, going both ways, for a specific individual contact.   Is this possible? Blocking any usage of iMessage between myself and this other individual, so they must either send and receive any texts to or from me via SMS (or the texts fail/never arrive). Biblatex ieee incorrect display of volume and issue number How to output MySQL query results in CSV format? Prove that $4\sin\frac{2\pi}{7}- \tan\frac{\pi}{7}= \sqrt{7}$ . Prove that $$4\sin\frac{2\pi}{7}- \tan\frac{\pi}{7}= \sqrt{7}$$   I think midway calculations are not easy formulas, I can't find what kind of formula transformation to solve! I think the best solution here is using right triangle . . . I have one solution too, but not pretty . ! Why can't we see infrared light? Printing pointers in C I was trying to understand something with pointers, so I wrote this code:  #include &lt;stdio.h&gt;  int main(void) {     char s[] = "asd";     char **p = &amp;s;      printf("The value of s is: %p\n", s);     printf("The direction of s is: %p\n", &amp;s);      printf("The value of p is: %p\n", p);     printf("The direction of p is: %p\n", &amp;p);      printf("The direction of s[0] is: %p\n", &amp;s[0]);     printf("The direction of s[1] is: %p\n", &amp;s[1]);     printf("The direction of s[2] is: %p\n", &amp;s[2]);      return 0; }   When compiling it with gcc I get these warnings:  $ gcc main.c -o main-bin -ansi -pedantic -Wall -lm main.c: In function ‘main’: main.c:6: warning: initialization from incompatible pointer type main.c:9: warning: format ‘%p’ expects type ‘void *’, but argument 2 has type ‘char (*)[4]’ main.c:11: warning: format ‘%p’ expects type ‘void *’, but argument 2 has type ‘char **’ main.c:12: warning: format ‘%p’ expects type ‘void *’, but argument 2 has type ‘char ***’   (The flags for gcc are because I must be C89)  Why incompatible types of pointer? Isn't the name of an array a pointer to it's first element? So if s is a pointer to 'a', &amp;s must be a char **, no? And why do I get the other warnings? Do I have to cast the pointers with (void *) in order to print them?   And when running I get something like this:  $ ./main-bin The value of s is: 0xbfb7c860 The direction of s is: 0xbfb7c860 The value of p is: 0xbfb7c860 The direction of p is: 0xbfb7c85c The direction of s[0] is: 0xbfb7c860 The direction of s[1] is: 0xbfb7c861 The direction of s[2] is: 0xbfb7c862   How can the value of s and it's direction (and of course the value of p) be the same? Is it possible to log back into a Pokemon Go account? If we disbelieve $H_0$, why quote a p value calculated assuming $H_0$ was true? An idiom to describe someone who thinks he/she is wiser than others Struggling with Plural/Singular? Which of the following is a grammatically correct statement?   The value of cars depreciate over time.   The values of cars depreciate over time. The value of cars depreciates over time.   Thanks! Exercise 3.5.12. Let $ f: \mathbb N \times \mathbb N \to \mathbb N$ be a function, and let c be a natural number. Show that there exists a function $a:\mathbb N\to \mathbb N$ such that $$a(0)=c$$ and $$a(n++)=f(n,a(n))\;for\;all\;n\in \mathbb N,$$ and furthermore that this function is unique. Tao gives the hint: first show inductively, by a modification of the proof of Lemma 3.5.12, that for every natural number $N\in \mathbb N$, there exists a unique function $a_N:\{ n\in \mathbb N:n\lt N\}\to \mathbb N $ such that $a_N(0)=c$ and $ a_N(n++)=f(n,a(n))$ for all $n\in \mathbb N$ such that $n\lt N$.  In my opinion, we could solve it by proving that $\forall n\in \mathbb N,a(n)\in \mathbb N$ according to mathematcial induction.  The property P(n) is:$$\forall n\in \mathbb N, a(n)\;is\;unique$$ P(0) is true. And if P(n) holds, P(n++) is true because different natural numbers must have different successors. $$a(n++)=f(n,a(n))$$ $$a(m++)=f(m,a(m))$$ $$n++=m++\Rightarrow n=m\Rightarrow a(n++)=a(m++)$$ It's similar to the proof of  Proposition 2.1.16 (Recursive definitions): Suppose for each natural number $n$, we have some function $f_{n} : \mathbb{N} \rightarrow \mathbb{N}$ from the natural numbers to the natural numbers. Let $c$ be a natural number. Then we can assign a unique natural number $a_{n}$ to each natural number $n$, such that $a_{0} = c$ and $a_{n{++}} = f_{n} (a_{n})$ for each natural number $n$.  This proof indicates both the existence and uniqueness of the function $a$. I don't know why Tao says it's informal.  But regarding the hint, how do we get the function f even if we've got the conclusion from the hint. Also it seems like the function a(n) can be diffentent considering different N. I don't know if the hint is correct.