f: $\mathbb{R}$ $\to$ $\mathbb{R}$ is a continuous function and $\overline{f(\mathbb{N})} = \mathbb{R}$. Prove that f is surjective. I've thought alot about this but i still haven't find a clue. I would really appreciate any help or a hint.
Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous map such that $\overline{f(\mathbb{N})} = \mathbb{R}$. Prove that $f$ is surjective.  I tried to use the definition of the continuity and $\overline{f(\mathbb{N})} = \mathbb{R}$ to show that $f$ is surjective but it didn't work. Since $f$ is continuous, the preimage of every open set is open. Also it results in $f(\overline{A})\subseteq \overline{f(A)}$ for every set $A$. How this couples with $\overline{f(\mathbb{N})} = \mathbb{R}$ to show surjectivity?
How do I monitor network usage on a Windows server? I am very confused about the probability formula $E[\min(X,k)]=\int_0^kxf(x)dx+k(1-F(k))$ I understand the proof of this first formula.  I am confused about where he got the first E[min(X, k)] formula. It seems to have just come out of nowhere. And also, can someone explain why the two formulas are correct please? As far as I know, ping needs to create a  (which needs either root access or cap_net_raw capabilities). From my understanding the trend these last years has been to remove  binaries and replaced them with . However when I look at the ping binary on my Fedora 32, it doesn't look to have any: $ ls -la $(which ping) -rwxr-xr-x. 1 root root 82960 May 18 10:26 /usr/bin/ping $ sudo getcap -v $(which ping) /usr/bin/ping $     Does ping need to open raw socket on fedora? Or is there another way to give it the permission to open a raw socket? 'Guest' vs 'Guest User License' in sites? What's the preferred method of writing AJAX-enabled plugins? Does the alternating sum of prime reciprocals converge? $$\sum_{n = 1}^\infty \frac{(-1)^n}{p_n}$$ where $p_n$ is the $n$th prime.  I have computed this to 10000 rather than infinity. My results suggest that convergence does happen but it's very slow. But I can't even be sure about the first few digits: $-0.26959$?  I have looked at the "questions that may already have your answer" and some of the "similar questions," but they involve somewhat different formulas. Where are the default settings saved for the Command Prompt defaults?  Are they stored in the registry somewhere, or in a file?  I would love to make a .reg file to set my default preferences when logging into a machine for the first time.  I'm talking about the default application settings: QuickEdit mode, buffer size, text color, etc. Why do I have two docks in Ubuntu desktop (with GNOME 3) after installing Dash to Dock? What's going on here? I've installed the  on Ubuntu 17.10. Everything was cool, and as I was trying to hone and tweak the look of my desktop, at some point the shortcuts bar (dock?) on the left is duplicated.  (Click image to enlarge) The default, which normally disappears with Dash to Dock, is there underneath the one that usually shows up when I install Dash to Dock. Why? I've tried to go through the options for Dash to Dock, the GNOME Tweak Tool, and the regular Ubuntu settings, but I can't figure it out. In "Knots and Primes: An Introduction to Arithmetic Topology", the author uses the following proposition   Let $h: Y \to X$ be a covering. For any path $\gamma : [0,1] \to X$ and any $y \in h^{-1}(x) (x = \gamma(0))$, there exists a unique lift $\hat{\gamma} : [0,1] \to Y$ of $\gamma$ with $\hat{\gamma}(0) = y$. Furthermore, for any homotopy $\gamma_t (t \in [0,1])$ of $\gamma$ with $\gamma_t = \gamma(0)$ and $\gamma_t(1) = \gamma(1)$, there exists a unique lift of $\hat{\gamma_t}$ such that $\hat{\gamma_t}$ is the homotopy of $\hat{\gamma}$ with $\hat{\gamma_t}(0) = \hat{\gamma}(0)$ and $\hat{\gamma_t}(1) = \hat{\gamma}(1)$.  The author then follows with; "In the following, we assume that any covering space is connected. By the preceding proposition, the cardinality of the fiber $h^{-1}(x)$ is independent of $x \in X$."   I am not sure why this results is true. This is my attempt at explaining it to myself. Take two different $x_1, x_2$ and their fibers $h^{-1}(x_1), h^{-1}(x_2)$ such that $y_1 \in h^{-1}(x_1)$ and $y_2 \in h^{-1}(x_2)$. Take two paths $\gamma_1, \gamma_2$ such that $\gamma_1(0) = x_1$ and $\gamma_2(0) = x_2$. Then we get two lifts $\hat{\gamma_1}, \hat{\gamma_2}$ with $\hat{\gamma_1}(0) = y_1$ and $\hat{\gamma_2}(0) = y_2$. Now, since our covering space is connected we can continuously deform $\hat{\gamma_1}$ into $\hat{\gamma_2}$ and conclude that every elements in $h^{-1}(x_1)$ is also in $h^{-1}(x_2)$ and vice versa. I feel that this is wrong but cannot figure out the right way to see this. Any help would be appreciated. How can I have  for Ubuntu? If I install the Dev Preview 3 on my Nexus 5 right now, will I get an OTA to the stable build or will I have to flash it as well? Does photon have size measurement because of its particle nature Does photon have size measurement because of its particle nature like electron's 3.86*10^-13m etc.. Theorem. If the identity is written as the product of $r$ transpositions, $id=τ_1τ_2\dots τ_r$, then $r$ is an even number. Proof. We will employ induction on $r$. A transposition cannot be the identity; hence, $r &gt; 1$. If $r = 2$, then we are done. Suppose that $r &gt; 2$. In this case the product of the last two transpositions, $τ_{r−1}τ_r$, must be one of the following cases: $(ab)(ab) = id\\ (bc)(ab) = (ac)(bc)\\ (cd)(ab) = (ab)(cd)\\ (ac)(ab) = (ab)(bc),$ where $a$, $b$, $c$, and $d$ are distinct. The first equation simply says that a transposition is its own inverse. If this case occurs, delete $τ_{r−1} τ_r$ from the product to obtain $id=τ_1τ_2···τ_{r−3}τ_{r−2}$. By induction $r − 2$ is even; hence, $r$ must be even. In each of the other three cases, we can replace $τ_{r−1} τ_r$ with the right-hand side of the corresponding equation to obtain a new product of $r$ transpositions for the identity. In this new product the last occurrence of $a$ will be in the next-to-the-last transposition. We can continue this process with $τ_{r−2} τ_{r−1}$ to obtain either a product of $r − 2$ transpositions or a new product of $r$ transpositions where the last occurrence of $a$ is in $τ_{r−2}$. If the identity is the product of $r − 2$ transpositions, then again we are done, by our induction hypothesis; otherwise, we will repeat the procedure with $τ_{r−3}τ_{r−2}$. At some point either we will have two adjacent, identical transpositions canceling each other out or $a$ will be shuffled so that it will appear only in the first transposition. However, the latter case cannot occur, because the identity would not fix $a$ in this instance. Therefore, the identity permutation must be the product of $r − 2$ transpositions and, again by our induction hypothesis, we are done. Question: That is the proof written in the book, but unfortunately after more than $10$ times reading I don't understand the last two paragraph (i.e., from &quot;In each of the other three cases&quot;). Is there another simpler proof or could someone please clarify what is going on? Thank you. Why does centering not work with calligra? $f$ is surjective iff it has a right inverse: using the axiom of choice and errors in ProofWiki Jewish Law prohibits the  However, butter, a milk product, is a kitchen staple for most cooked emulsions (eg. pan-sauces, sauces béchamels, or roux-based sauces). While I use appropriate cooking fats for making non-dairy versions for meat dishes, this frequently doesn't work.  What is a non-dairy fat which may be used to as an emulsifier or roux ingredient? How can I test whether my camera is working after it was dropped? Issue : How can I evaluate my cameras performance synthetically?  Scenario : Bought a Nikon D3100 an year back. Photos looked good enough. The camera fell, zoom lens effected.  But works. Cleaned lens using soap, water etc.  Carefully. Now photos look much worse.  How do I know if Photos are really bad?  Is there a tool?    Sample Photo  Now (Bad)   Original (Good)   Please help me realize whether my camera has gone bad.  Can you tell your personal opinion also? Do I need a UK or Ireland transit visa when flying from the US to a Schengen country? Finite group for which $|\{x:x^m=e\}|\leq m$ for all $m$ is cyclic. Let $G$ be a finite group. For each positive integer $m$, if $x^{m}=e$ has at most $m$ solutions in $G$, $G$ is cyclic.   What I have thought is that $n=\sum_{d\mid n}\phi(d)$ can be used to solve this and showing that $|G|$ order element in $G$ exists is enough. I have a small issue with configuring the lidR  for parametrizing the extracted raster files with grid_metrics() function.  My question is how to set in the catalog options that:   be able to overwrite the results with a second run if I want to run my script several times and I do not want to name each output differently. instead of .tif I would like to get .grd files as output.   My current code (which is working) is the following:  normalizedctg = catalog(workingdirectory) opt_output_files(normalizedctg) &lt;- paste(workingdirectory,"/clipped/normalized/{XLEFT}_{YBOTTOM}_height",sep="") height = grid_metrics(normalizedctg, max(Z),res = resolution)   But with a second run on the same code I get the following error:  Error in .getGDALtransient(x, filename = filename, options = options,  :    filename exists; use overwrite=TRUE