Show that a continuous function doesn't exist such that for every real $c$, there exist exactly two reals $a, b$ s.t. $f(a)=f(b)=c$. I got pretty confused at that question... Can anyone help me understand this question better and/or drop some hints please? Thank you!
Show that there's no continuous function that takes each of its values $f(x)$ exactly twice. I need to prove the following:  There's no continuous function $f:[a,b]\to \mathbb{R}$ that takes each of its values $f(x)$, $x\in [a,b]$  exactly twice.  First of all, I didn't understand the question. For example $x^2$ takes $1$ twice, in the interval $[-1,1]$. Is it saying that it does not occur for all $x$ in the interval? But what about $f(x) = c$? Is it saying that it does not occur only exactly $2$ times, then? I have no idea about how to prove it. I know that for $f(x)$ such that $f(a)&lt;f(x)&lt;f(b)$, if $f$ is continuous then there is a $c\in [a,b]$ such that $f(c) = f(x)$.  Now, there's the following proof in my book and I really wanted to understand it, instead of just getting a new proof  Since the interval $[a,b]$ has only $2$ extreme points, then the maximum or minimum of $f$ must be in a point $c\in int([a,b])$ and and in another point $d\in [a,b]$. Then, there exists $\delta&gt;0$ such that in the intervals $[c-\delta, c), (c,c+\delta)$ (and if $d$ is not extreme of $[a,b]$, $[d-\delta, d]$) the function takes values that are less than $f(c) = f(d)$. Let $A$ be the greatest of the numbers $f(c-\delta), f(c+\delta), f(d-\delta)$. By the intermediate value theorem, there are $x\in [c-\delta, c), y\in (c, c+\delta]$ and $z\in [d-\delta, d)$ such that $f(x)=f(y)=f(z)=A$. Contradiction.   Well, why the last part? Why is it that I can apply the intermediate value theorem to these values? For example, $&lt;f(c-\delta)&lt;p&lt;f(c)$, then by the theorem I know that there exists $m\in [c-\delta, c)$ such that $f(m) = p$. Same for the other intervals. But what guarantees thhat the greatest of the values between  $x\in [c-\delta, c), y\in (c, c+\delta]$ will be inside the intervals $[c-\delta), c), (c,c+\delta), [d-\delta, d)$?
How can I set the location (as it's picked up in CoreLocation services) in the iPhone Simulator? Limits of dead reckoning using MEMS sensors I'm trying to track body parts relative to a person's torso. I see quite a few questions about using MEMS accelerometers and gyros for dead reckoning, and they confirm my suspicions that various factors greatly limit their usefulness for these sorts of applications, but I'm seeking clarification of these limits:   What exactly are these limits?  Other answers have addressed why these limits exist. Naturally the specifications the parts in the system in question and what is considered "acceptable error" for the system will both change the exact limits, but is there a single order of magnitude in time, or distance that I can expect dead reckoning to work? I'm well aware that over long distances (a few yards or so) the error becomes too large for most practical purposes, but what about within a few feet?  What can I do to improve these limits?  I'm currently looking at using an accelerometer and a gyro. What other sensors can I add to the system to improve the error rate? I know over longer distances a GPS can be used, but I doubt any consumer electronics grade GPS has fine enough resolution to help in my case.  Additionally, a general consensus seems to the only way to improve these limits past the point of improved sensors is to provide a reference not subject to error. Some systems solve this using cameras and markers.  What kind of reference points can a portable/wearable device provide?   I've seen the usage of radio waves to measure long distances accurately, but I can't tell if such a system could be accurate on such small scale (in terms of distance measured) using "off-the-shelf" components. Closed formula for the sums $\sum\limits_{1 \le i_1 < i_2 < \dots < i_k \le n} i_1 i_2 \cdots i_k $? I've worked out a few summation formulas, and I am hoping to find a pattern. Unless I have made a mistake somewhere, we have the following identities:  $$\sum_{1 \le i \le n} i = \frac{(n+1)n}{2} $$  $$\sum_{1 \le i &lt; j \le n} ij = \frac{(3n+2)(n+1) \, n \, ( n-1)}{24}$$   $$\sum_{1 \le i &lt; j &lt; k \le n} ijk = \frac{(n+1)^2 \, n^2 \, (n-1)(n-2) }{48}$$   It seems clear that $$p_k(n)= \sum_{1 \le i_1 &lt; i_2 &lt; \dots &lt; i_k \le n} i_1 i_2 \cdots i_k $$ is a polynomial in $n$ of degree $2k$. But is there a nice closed-form formula for it? Maybe in terms of its factorization?  Comment: I realize that what I am looking for is the coefficient of $t^k$ in the expansion $$(1+t)(1+2t) \dots (1+nt),$$ so maybe generating function techniques could be helpful. But the main way I know to extract the coefficient of $t^k$ is to take derivatives $k-1$ times and on the surface of things that looks like a huge mess. Is it possible to set DNS for mobile network Is "has or will read" grammatical? Are quotes needed for local variable assignment? Can I safely omit quotes on the right side of a local assignment?  function foo {     local myvar=${bar}     stuff() }   I'm mainly interested in bash, but any info on corner cases in other shells are welcome. Is there a way to get the raw SQL string executed when calling PDOStatement::execute() on a prepared statement? For debugging purposes this would be extremely useful. Why do I need an extended partition? What is an extended partition? Why do I need to use it to install Ubuntu if I already have four partitions that are factory installed? Solutions of $p!q! = r!$ The title says it all, more or less. Obviously, there are infinitely many "trivial" integral solutions of the form $p=n, q=(n!-1), r= n!$. How many non-trivial solutions are there?  I came across this about ten years ago; as far as I can tell, it hasn't appeared here before, so I thought that it might be of interest. I'm actually most interested in finding whether there was any progress made since While trying some problems along with my friends we had difficulty in this question.   True or False: The value of the infinite product $$\prod\limits_{n=2}^{\infty} \biggl(1-\frac{1}{n^{2}}\biggr)$$ is $1$.   I couldn't do it and my friend justified it by saying that since the terms in the product have values less than $1$, so the value of the product can never be $1$. I don't know whether this justification is correct or not. But i referred to Tom Apostol's Mathematical Analysis book and found a theorem which states, that    The infinite product $\prod(1-a_{n})$ converges if the series $\sum a_{n}$ converges.   This assures that the above product converges. Could anyone help me in finding out where it converges to? And,   Does there exist a function $f$ in $\mathbb{N}$ ( like $n^{2}$, $n^{3}$) such that $\displaystyle \prod\limits_{n=1}^{\infty} \Bigl(1-\frac{1}{f(n)}\Bigr)$ has the value $1$? How to hide "internal" polygon boundaries? How can I find the files that were stored on my flash drive before I installed Ubuntu to it?   Edit: No, I didn't format it, my files are still on the flash drive root. I just need to know how to access them there. Would like to emulate google maps' driving directions functionality but use pgRouting instead. google's data over the project area &amp; scale is quite a bit off so we want to use our own transport network data in postGIS.  any hints?  thanks. Prove the map sending a matrix to it's inverse is continuous. Let $GL(n,\mathbb{R})$ be the space of invertible $n \times n$ matrices over $\mathbb{R}$. I have already proved that $GL(n,\mathbb{R})$ is open in $\mathbb{R}^{n^2}$ and that it is not connected with the relative topology from $\mathbb{R}^{n^2}$. How do I prove that the map $f:GL(n,\mathbb{R})\rightarrow GL(n,\mathbb{R})$ given by $f(x):=x^{-1}$ is continuous?  Any help would be greatly apprecaited.   Additionally how do I prove that $g:GL(n,\mathbb{R})\times GL(n,\mathbb{R}) \rightarrow GL(n,\mathbb{R})$ given by $g(x,y):=xy$ (matrix multiplication) is continuous, where we take the relative topology from $\mathbb{R}^{n^2} \times \mathbb{R}^{n^2} \cong \mathbb{R}^{2n^2}$? How to calculate $\lim_{n\to\infty}(1+1/n^2)(1+2/n^2)\cdots(1+n/n^2)$? How can we compute the following limit:     $$\lim_{n\to\infty}(1+1/n^2)(1+2/n^2)\cdots(1+n/n^2)$$   Mathematica gives the answer $\sqrt{e}$. However, I do not know to do it. Retrieve globalpicklist/globalvalueset without SOQL Is it possible to retrieve the values of a Global Picklist/GlobalValueSet using Schema class without a SOQL ? Selling a car with a lien I own a car personally, but it is mortgaged to a bank. I want to sell it to an individual. I haven't the funds to repay the loan until I'm paid for the car. The buyer doesn't want to pay me without a lien release (or guarantee that one is coming). The bank won't release the lien until it gets the funds. What do people do in a case like this? It must be a common case. What should I do? In a bash script what does a dot followed by a space and then a path mean? I came across this example when trying to mount a usb device inside a openvz container and I have never seen the construct in the second line before. Can you explain what it signifies?  #!/bin/bash . /etc/vz/vz.conf I aim to show that $$(n+1)^n&lt;n^{(n+1)}$$ for all $n \geq 3$. I tried induction, but it didn't work. What should I do? What're the differences between PCA and autoencoder? Both PCA and autoencoder can do demension reduction, so what are the difference between them? In what situation I should use one over another?