How can I prove that $f(x)=x^s$ for $0&lt; s ≤1$ is Hölder-continuous with constant s?
I want to show that $$f: \mathbb{R}_{\geq0} \to \mathbb{R}_{\geq0}$$ $$x \mapsto x^s$$ is Holder continuous with Holder exponent $s \in \mathbb{R}$, where $0&lt;s \leq 1$. So what I want to show is that $\exists \hspace{2 mm} C \in \mathbb{R}_{\geq0}$ sucht that for all $ x,y \in \mathbb{R}_{\geq0},  $ $$|x^s -y^s| \leq C|x-y|^s$$ and therefore, assuming, wlog $\hspace{1mm} x&gt;y$ $$(x^s -y^s) \leq C(x-y)^s.$$  I thought about Bernoulli's inequality but couldn't make that work.I thought about the binomial theorem, but didn't know how to handle the fact that $s \in \mathbb{R}$.
I am wondering if I can minimize a multi-variable function one variable at a time. In other words, is it true that:  $min_{x_1,x_2} f(x_1,x_2)=min_{x_1} min_{x_2} f(x_1,x_2)$ I used osm2pgsql to import data into my PostgreSQL database and it put gave me the following 4 tables, planet_osm_point, planet_osm_line, planet_osm_polygon and planet_osm_roads.   I have looked at the schema for osm2pgsql and inside the planet_osm_point table there is a column 'way' of type point. I tried to query it as an array like it says in the documentation here ():     It is possible to access the two component numbers of a point as   though the point were an array with indexes 0 and 1. For example, if   t.p is a point column then SELECT p[0] FROM t retrieves the X   coordinate and UPDATE t SET p[1] = ... changes the Y coordinate. In   the same way, a value of type box or lseg can be treated as an array   of two point values.   Anyone know the correct query the planet_osm tables from osm2pgsql based on long/lat? For example if I wanted to retrieve the information for Big Ben (London) I would query along the lines of:      SELECT * FROM planet_osm_point      WHERE ST_Y(ST_Transform(way, 4326)) = '-0.12'        AND ST_X(ST_Transform(way, 4326)) = '51.5';   There is no error message, it comes out with the table names, but no data is output either. (Its an empty table). What am I doing wrong here?  Perhaps it is too precise, i.e. there are no coordinates that match it exactly? Perhaps I need to try being more general, say 51 &lt; long &lt; 52 and -0.5 &lt; lat &lt; 0. Although this would throw back multiple results so would not be quite as accurate. The derivation of the Hamilton-Jacobi equation using canonical transformations is typically done involving a type-2 generating function.  Is it possible to use a another type of generating function, namely type 1, 3 or 4? Show that $[0, 1)\times[0, 1)$ is homeomorphic to $[0, 1]\times[0, 1)$ but not to $[0, 1]\times[0, 1]$. When I sketch these spaces it this statement makes sense intuitively because $[0, 1]\times[0, 1]$ is closed and $[0, 1)\times[0, 1)$ and $[0, 1]\times[0, 1)$ are each "missing" part of their boundary which would make them closed. I also know, $[0, 1]\times[0, 1]$ is compact and compact spaces are not homeomorphic to non-compact spaces.   I am having trouble saying this explicitly, as is pretty obvious in the clumsiness of the wording above. Given a sample of n units out of a population of N, population median can be estimated by the sample median.  How can we get the variance of this estimator? I'm planning to visit a summer school in the Schengen zone. What kind of visa should I apply for? I will be attending a summer school which will last 10 days.  I want to know: for applying for a Schengen visa, is the purpose of my trip considered as touristic or study? How prove there is no continuous functions $f:[0,1]\to \mathbb R$, such that $f(x)+f(x^2)=x$. Prove that there is no continuous functions $f:[0,1]\to R$, such that   $$ f(x)+f(x^2)=x. $$   My try. Assume that there is a continuous function with this property. Thus, for any $n\ge 1$ and all $x\in [0,1]$, \begin{align*} f(x)&amp;=x-f(x^2)=x-\big(x^2-f(x^4)\big)=x-x^2+\big(x^4-f(x^8)\big)=\cdots\\ &amp;=x-x^2+x^4-\cdots+(-1)^n\left(x^{2^n}-f\big(x^{2^{n+1}}\big)\right) \end{align*}  since $f(0)=0$ and $\displaystyle\lim_{n\to \infty}x^{2^{n+1}}=0$ for any $x\in(0,1)$,it follows by the continuity of $f$ that $\displaystyle\lim_{n\to \infty}f\big(x^{2^{n+1}}\big)=0$, hence $$f(x)=x-x^2+x^4-x^8+\cdots+(-1)^nx^{2^n}+\cdots$$ for any $x\in (0,1)$  Why do I have prove that there is exists such a continuous functions $f$? Maybe  my example is wrong? Why?  if my   method is wrong,then How prove this problem ? Thank you. See dashboard as non logged user in community Is it possible, as a non logged user, to see report and dashboard in community when it is open to non logged user? And  for analytic dashboards? I'm taking Calculus II next semester so I don't have all the tools from it to back me up but something came up in a discussion this evening that asked me to prove that the following sequence converges. As I'm new to formal math I'm hoping someone can verify my proof, especially step $6$. I must prove that $$\lim_{n\to\infty}P_n=b$$ where $$P_{n+1} = P_n + \frac{b-P_n}{2}$$ $$P_0 = a$$ and $a,b \in \mathbb{R}$ such that $a&gt;b$ My proof goes as follows:   We must prove that $$\lim_{n\to\infty}P_n=b$$  That is, $\forall\epsilon&gt;0$ there exists a $\delta$ such that there exists an $n$ satisfying $$n&gt;\delta \iff b-P_n&lt;\epsilon$$  First, let us prove that $P_n$ is bounded by $b$ by contradiction. Let us choose an $i$ such that $P_i &lt; b$ and $P_{i+1}&gt;b$. Then we have $P_{i+1} = P_i + \Delta P$ where $\Delta P &gt; b-P_i$. but this is impossible because by the definition of the recurrence $\Delta P = \frac{b-P_i}{2} &lt; b-P_i$. Therefore, $P_i$ is bounded from above by $b$.  Additionally, it is trivial to show that $P_{i+1} &gt; P_i$.  With this in mind, choose an arbitrary $\epsilon$. We need to show that there exists an $n$ such that $P_n &gt; b-\epsilon$. Assume the contrary, that $P_n$ is bounded by $\epsilon$.  Using a similar argument to what which was used before, assume $P_n$ is bounded by $\epsilon$ and then there exists a $P_i$ such that $P_{i+1}\not &gt; P_i$. But this results in a contradiction with (4)  Therefore, $P_n$ is not bounded by $\epsilon$ and we have proved what we want to prove. I added some extra repositories with the Software Sources program. But when I reload the package database, I get an error like the following:     W: GPG error:  trusty InRelease: The following signatures couldn't be verified because the public key is not available: NO_PUBKEY 8BAF9A6F   I know I can fix it using apt-key in a terminal, according to the official Ubuntu documentation. But I would have liked to do it graphically. Is there a way to do this without using a terminal? If you create a scene for the Video Sequence Editor and start working on a project Blender will render the output from the VSE. But if you want to later switch it to render your 3D scene again (the camera output) where is the setting to select the source?  Usually I keep a separate .blend for VSE edits so I don't need to switch, but I feel like I ought to learn how to do this as it's pretty basic. I would expect to find this in the Render context but I see no place to specify the source.  Thanks. Login with Facebook using Stack Exchange Android App SSH: How to disable weak ciphers? How to install LINE (messaging app) on Ubuntu I would like to install  chat platform on my Ubuntu 12.04 LTS system. Please, could anyone provide me the steps to successfully install this software?. I've been trying to install this software using wine but I couldn't succeed it. AmpScript to select Right or Left characters from a string Is there an AmpScript function that can be used in an equivalent manner to sql left, right and mid functions?   Thanks, Mark How do I deal with a runaway bride? Proving an infinite number of primes of the form 6n+1 The proofs given on other sites weren't that clear and used different methods that I have yet to learn.    Prove that there are an infinite number of primes of the form 6n+1.  The hint that was given was: Let p = p1, p2, ..., pk + 1, where p1 = 2, p2 = 3,...pk are the first k primes.  Show that p is prime.  (p1 means p sub 1, p2 means p sub 2, and pk is p sub k.  Wasn't sure how to write it on this.)    Can someone explain this hint on how they came about of p1 = 2, p2=3, etc, and prove this please?    Also, how would the proof change if the form changed?  ("Prove that there are infinite number of primes of the form....") I had the Facebook and Instagram apps installed on my iPhone (latest iOS). I deleted both apps and checked that there was no data from either app stored on iCloud (I went to Settings - iCloud - Manage storage and there was no mention of them storing anything on iCloud). Then I reinstalled Instagram, opened it, clicked on Login and the page still showed “Login as MYFACEBOOKNAME”.   How is this possible? How can I delete my FB login info once and for all on my iPhone? Show that $\sqrt{n^2+1}-n$ converges to 0 I want to use the definition of the limit to show that $\sqrt{n^2+1}-n$ converges to 0.  The definition is as follows: if $\sqrt{n^2+1}-n$ converges to 0, then $\forall \epsilon&gt;0$, there exists an $N&gt;0$ such that $n\ge N \implies \mid\sqrt{n^2+1}-n\mid&lt;\epsilon$.  Now I want to start backwords in order to figure out how to pick N. I know:   $\mid\sqrt{n^2+1}-n\mid=\sqrt{n^2+1}-n$   since $n$ is a natural number and $\sqrt{n^2+1}&gt;n$. So I need to pick an N such that $\sqrt{n^2+1}-n&lt;\epsilon$. I tried multiplying $\sqrt{n^2+1}-n$ by $\frac{\sqrt{n^2+1}+n}{\sqrt{n^2+1}+n}$ but that didn't really seem to help.   Do you have any ideas on how to find this N?  Thanks I was wondering if there is a viable place on Stack Exchange to ask economics-related questions.  Questions I'd have in particular could be equation-based or theoretical.  I suppose the equation-based questions could possibly fit into mathematics but I haven't checked their FAQ.