OpenStreetMap - only administrative regions
Seeking world map with boundaries of regions and communities for commercial use?
Increasing real valued function whose image set is connected Let $S = [0,1) \cup [2,3]$ and $f\colon S \rightarrow \mathbb R$ be such that $f(S)$ is connected . Which of the following are true:  a) $f$ is discontinuous exactly at one point.  b) $f$ is discontinuous exactly at two points.  c) $f$ is discontinuous at infinite many points.  d) $f$ is continous.  I think $f$ is continuous . We need to show that $\lim_{x \to a^+}f(x) = f(a)$ for any $a \in S$. Since $f(x)$ is increasing, $\lim_{x \to a^+}f(x) \leq f(a)$. Suppose $\lim_{x \to a^+}f(x) &lt; f(a)$. Then there is $l$ such that $\lim_{x \to a^+}f(x) &lt; l &lt;f(a)$, which contradicts connectedness of $f(S)$. Similarly we can show that $\lim_{x \to a^-}f(x) = f(a)$.  I would be thankful if anyone would check my solution and correct me if there are any errors. I was reading a blogpost here:  () One thing that threw me off was that after the first four large displayed equations, there is the statement &quot;it is sufficient to prove that if the theorem holds for $n=m+1$, then it holds for $n=m$.&quot; How is this type of induction valid? I browsed around for things like backward induction, reverse induction, and Cauchy induction, but couldn't find a justification for how this is valid. With the usual forward induction of verifying a base case and proving $P(n)\implies P(n+1)$, it's easy to intuitively understand how induction will show a property holds for all natural numbers (or at least starting at the base case). But with this reverse induction, it seems to me that if you prove $P(m+1)\implies P(m)$, then if you were able to verify a specific case like $P(15)$, then you would only know it's true for numbers up to $15$. How does it actually prove the property for all naturals? Adding boolean yes/no field in ArcGIS Desktop? How does one add a field to an ArcGIS feature class with a boolean data type? That is an attribute where the allowed value is only one of a pair such as 1/0, Yes/No, On/Off, Present/Not-present, etc.   A boolean data type is not available in the Data Type pick-list. When quoting something said by someone else, should I use a comma before the quote? Suppose I quote the definition given in a dictionary. Am I correctly using the the punctuation marks in the following sentences? (I am using them in the way I think correct in American English.)     One of the meanings of cat is, "a small animal with soft fur that people often keep as a pet."        A cadet is, "a young person who is training to become an officer in the police or armed forces."   Should I use a comma only when I explicitly quote somebody?     Groucho Marx once said, "Although it is generally known, I think it's about time to announce that I was born at a very early age."   Should I use a comma in this case too?     As Groucho Marx said,            No one is completely unhappy at the failure of his best friend.      On Comma Sense, A fun-damental guide to punctuation (Richard Lederer and John Shore), I read the following about using the comma:     Use commas to set off complete quotations:           The great general George S. Patton once said, "No, no—the war is this way, you idiots!" NOT IN vs NOT EXISTS Evaluate $\lim_{n\rightarrow\infty} \int_0^1 f(x^n) dx$ Let $f$ be a real, continuous function defined for all $0\leq x\leq 1$ such that $f(0)=1$, $f(1/2)=2$, and $f(1)=3$. Show that  $$\lim_{n\rightarrow\infty} \int_0^1 f(x^n) dx$$  exists and compute the limit.  Attempt:   Since $f$ is real-valued continuous on $0\leq x\leq 1$ and the boundaries $f(0)=1$ and $f(1)=3$, $f$ is bounded on the interval  and the integral $\displaystyle\int_0^1 f(x^n)dx$ exists for any positive $n$. Thus, we can interchange the limit and the integral and compose the limit, \begin{align*} \lim_{n\rightarrow\infty} \int_0^1 f(x^n)dx&amp;=\int_0^1\lim_{n\rightarrow\infty}f(x^n)dx\\ &amp;=\int_0^1 f\left(\lim_{n\rightarrow\infty} x^n \right)dx\\ &amp;=\int_0^1 f(0)dx=1. \end{align*}  Questions: So, my first question concerns whether interchanging the limit and the integral is correct. It seems it would be justified by the dominated convergence theorem, where $f(x^n)$ is dominated by a function $g(x)=\displaystyle\sup_{0\leq x\leq 1} f(x^n)$ and $f_n(x^n)$, on $0\leq x\leq 1$, converges pointwise to a function $f$ that takes value 1 for $x\neq 1$ and 3 for $x=1$.  My next question is whether the interchange between the limit and composition of the function is allowed. Since the function is continuous and the limit exists at that point, it seems the interchange would be justified. (exercise from Tao's analysis book) Proof of a lemma relating to power set of X I'm stuck at one exercise from chapter of sets from Terence Tao's analysis book. I need to proof the lemma:  Lemma: Let $X$ be a set. Then the set $\{Y : Y \:\text{is a subset of}\: X\}$ is a set.  Note: The set $\{Y : Y \:\text{is a subset of}\: X\}$ is known as the power set of $X$, defined as  $2^X $  I can understand why the lemma is true, but I have no clue how to prove it. Furthermore, the author gave a hint that I found very confusing, it follows:  Hint: start with the set $\{0, 1\}^X$ and apply the replacement axiom, replacing each function $f$ with the object $f^{-1}(\{1\})$. Find a circle's radius with three known tangent lines I need to find the equation for a circle (mainly its radius) which is tangent to the following three lines:  $y = 0$  $y = \tan(70)x$  $y = -1.428148x + 0.790201$  For the last tangent line equation, I know that it is tangent at the point $(0.371272, 0.259968)$ However, for the other two I do not know the exact point of tangency, only that the circle is tangent to them. Text gets stretched or disappears when scrolling When scrolling in the Stack Exchange Android app, the text gets stretched vertically, code disappears, and sometimes the entire post content disappears.  This doesn't make the app completely useless, but it's pretty close.   Motorola Razr MAXX, Android 4.0.4. I've been trying to track a footage taken by my Samsung Galaxy Note 8.0 (GT-n5100) tablet. It has a good 5 mp camera and I tracked the footage well. But the problem arises at the time of solving the camera. The solve error is too high (more than 2). I don't know the width the tablet's camera sensor or the focal length or any other specifications. So my question is, has anybody been able to solve the camera on footage shot by a smartphone and if yes,  what camera and lense specs did you feed to the Blender? How do I force Python's print function to output to the screen? Why did Dumbledore drink the potion in the attempt to get a Horcrux? XeLaTeX update [20201004] and item ball in Beamer Beamer item balls go missing while compiling in XeLateX since yesterday's TeXlive update, however, the same code works with pdfLaTeX. Any solution, please. Here is my MWE: \documentclass{beamer}  \setbeamertemplate{itemize item}[ball] \setbeamertemplate{enumerate item}[ball]  \begin{document}  \begin{frame} \frametitle{A tested title} text \begin{enumerate}     \item this is an item \end{enumerate} text3 \begin{itemize}     \item this is an item \end{itemize} \end{frame} \end{document} Do Legendary Pokemon Respawn After The Elite Four? Unzip File: Permission Denied I am trying to unzip a file (Facebook download-your-info zip file), but here's what I got:   Is there a way I can force-unzip it?  Update: I tried sudo unzip file.zip -d destination_folder in terminal, and it unzipped the file, but what came out, all the files have size 0 bytes:   Update 2: I do have permission to the directory and the file.    (I'm really sorry if my question is asked badly. Let me know down in the comments :) ) Example where Hamiltonian $H \neq T+V=E$, but $E=T+V$ is conserved I'm looking for an example of a Hamiltonian $H$, where $H\neq T+V$, but the total energy in the system, $E=T+V$, is still conserved.   While I'm at it, I might as well add that I'd be most interested in an example from a classical field theory. Furthermore, I am looking for a nontrivial example, e.g. where $H$ doesn't just differ from $E$ by some gradient function that integrates to 0 in the action.   In particular, I am trying to get a better understanding of the physical implications of $H$ being conserved when nontrivial variables are used, i.e. physically, what does $H$ this correspond to. Conversely, can one have a system where $E = H$ in one frame, and is conserved, while in another frame $E'\neq H'$, and $H'$ is conserved while $E'$ is not? My international conference in Europe this May has been canceled due to the COVID-19. I have a ticket, purchased from TAP air Portugal, from Toronto, Canada to Lisbon. Because it is a non-refundable fare, they don't want to give a refund.  They won't allow me to reschedule later than May 31 of this year. This won't be useful for me.  What options do I have? I'd prefer to get a full refund.  (I am a US citizen if it matters)  Note: They offered to refund just the taxes, which is about 10% of the value. That's not horrible, but I'd rather have the whole thing returned. This question is also open on EL&amp;U:   Considering the amount of controversy it aroused, I believe it's past "Learners'" level.    An edit was suggested to my sentence.     There was were an orange, some grapes, two apples and a small pile of cherries on the plate.   In my native language plurality of the verb always follows plurality of the first element on the list. There were an orange,... sounds awkward to me, no matter what follows up. My simple solution was reordering:     There were some grapes, an orange, two apples and a small pile of cherries on the plate.   But that's not the first time I faced this situation and I'd like to know what the rules of grammar say about that — was my editor overzealous or am I trying to copy rules of my language that don't apply in English? Can you read my proof and tell me if it's correct? Thanks.  Let $V$ be a vector space over a complete topological field say $\mathbb R$ (or $\mathbb C$) with $\dim(V) = n$, base $e_i$ and norm $\|\cdot\|$. Let $v_k$ be a Cauchy sequence w.r.t. $\|\cdot\|$. Since any two norms on a finite dimensional space are equivalent, $\|\cdot\|$ is equivalent to the $l^1$-norm $\|\cdot\|_1$ which means that for some constant $C$, $\varepsilon &gt; 0$, $k,j$ large enough, $$ \varepsilon &gt; \|v_j - v_k\| \geq C \|v_j - v_k\|_1 e_i= C \sum_{i=1}^n |v_{ji} - v_{ki}| \geq |v_{ji} - v_{ki}|$$ for each $1 \leq i \leq n$. Hence $v_{ki}$ is a Cauchy sequence in $\mathbb R$ (or $\mathbb C$) for each $i$. $\mathbb R$ (or $\mathbb C$) is complete hence $v_i = \lim_{k \to \infty} v_{ki} $ is in $\mathbb R$ (or $\Bbb C$) for each $i$. Let $v = (v_1, \dots , v_n) = \sum_i v_i e_i$. Then $v$ is in $V$ and $\|v_k - v\| \to 0$:  Let $\varepsilon &gt; 0$. Then $$ \|v_k - v\| \leq C \|v_k - v\|_1 = C \sum_{i=1}^n |v_{ki} - v_i| \leq C^{'}n \varepsilon$$  for $k$ large enough. What is the simplest way of proving (to a non-mathematician) that the power set of the set of natural numbers has the same cardinality as the set of the real numbers, i.e. how to construct a bijection from $\mathcal{P}(\mathbb{N})$ to $\mathbb{R}$?