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Extending a continuous function defined on a subset of $\mathbb{R}$

Given $f:E \to \mathbb R $ continuous on $E$, define $\tilde f:\mathbb R \to \mathbb R $ as follows: For $\, x \in E\, $, let $\, \tilde f(x):=f(x)$. If $x \in \overline{E}\setminus E$ and $\...
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Prove that $\frac{a}{a^2+\lambda}+\frac{b}{b^2+\lambda}+\frac{c}{c^2+\lambda} \leq \frac{3}{\lambda +1}$

A comment on the tangent line method. Why does it work? The function $f$ is not concave everywhere. What's going on? The function $f_{\lambda}=f\colon x\mapsto \frac{x}{x^2 + \lambda}$ has derivatives ...
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3 votes

Prove that $\frac{a}{a^2+\lambda}+\frac{b}{b^2+\lambda}+\frac{c}{c^2+\lambda} \leq \frac{3}{\lambda +1}$

Tangent Line (TL) method: It suffices to prove that $$\frac{a}{a^2 + \lambda} \le \frac{1}{1 + \lambda} + \frac{\lambda - 1}{(1 + \lambda)^2}(a - 1). \tag{1}$$ (The desired result follows by summing ...
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Understanding the proof of the Poincaré's inequality

The right side of the inequality above (1.35) has the variables $x_2,\dotsc,x_d$ still free, while the right side has all of the variables $x_1,\dotsc,x_d$ still free. In this sense you can think of ...
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Prove that $\frac{a}{a^2+\lambda}+\frac{b}{b^2+\lambda}+\frac{c}{c^2+\lambda} \leq \frac{3}{\lambda +1}$

Define $$f(a,b,c,\epsilon)=\sum_{cyc}\frac{a}{a^2+\lambda}-\frac{1}{\lambda+1}-\epsilon(a+b+c-3)$$ $\frac{\partial}{\partial a}(\frac{a}{a^2+\lambda})=\epsilon=-\frac{a^2-\lambda}{(a^2+\lambda)^2}$for ...
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How to estimate difference of two normed elements?

If $a$ and $b$ are close to each other but their norms are small, multiplying them by the inverse of their norms can make the difference larger, for example: if $\|a\|=\|b\|=\delta$ then $$\left\|\...
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2 votes
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Confusion: Cannot understand the statement that $(0,1)^2$ cannot be represeted as union of countable open balls.

If a subset $U\subseteq \mathbb R^n$ is the union of (no matter how many, but at least two) disjoint open balls, then $U$ is necessarily disconnected. If $n=1$, then every bounded connected open ...
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How can I understand 'discontinuous on rationals' vs 'discontinuous on reals' intuitively?

Let's take the function whose value is 0 for every non-null real number, and 1 for 0. This function is continuous everywhere except on 0. So discontinuity on a single isolated point is possible. The ...
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Strong convergence does not implies weak star convergence

Counterexamples for a) and c). Let's take $E = c_0, E^* = l^1, E^{**} = l^\infty$. Let $e_n$ be the unit vectors in $l^1$, that is $e_n$ has coordinate $1$ in the $n$th position, zero elsewhere. Then:...
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1 vote

Proof that the sequence (0.9. 0.99. 0.999, . . . ) converges to 1. Is it correct?

This bit isn't correct: $|a_n -1|\leq |a_N-1|=|\frac{1}{10^n}|\leq|\frac{1}{10^N}|$ Most people can guess that you mean $|a_n -1|=|\frac{1}{10^n}|\leq|\frac{1}{10^N}|= |a_N-1|$ We take log base $10$...
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  • 5,953
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What makes $\frac{x(x+1)}{2}$ a "better" interpolant of the sum of the first $n$ positive integers than any other, and likewise in similar cases?

(Started as a comment, but became too long for one.) I don't know that there exists a universal "natural" criterion for such choices, but rather case-by-case rationales why a certain choice ...
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1 vote

How to remove a removable singularity

(i) What you did is exactly what I would have done. (ii/iii) To say something slightly less trivial, notice that if $f$ is holomorphic (indeed, even continuous) then $f(a) = \lim_{z \to a} f(z)$. You ...
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How can I approximate power consumption given the days billed in each month and the number of residents?

Start with the base calculation that you used. That is, you had a bill for $T$ amount, that comprised $D$ days, so the cost per day is $$\frac{T}{D}.$$ So, you have $D$ days, and each of the daily $\...
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2 votes

Question about $p$-adic absolute value

Here I use $|\cdot|$ as the usual p-adic absolute value. There is no $t>0$ such that $\varepsilon < t$ implies $|y| >\frac{1}{2}|x|$ when $|x-y|<\varepsilon$. If there were, we could pick ...
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  • 1,762
3 votes

$f(x)>0, f''(x)>0$. Prove: $\int_a^b f(x) dx > (b-a) f(\frac{a+b}{2})$

Let $L(x) = f'\left(\frac{a+b}{2}\right)\left(x - \frac{a+b}{2}\right) + f\left(\frac{a+b}{2}\right)$. Then, $L$ describes the line tangent to $f$ at $x=\frac{a+b}{2}$. Note that: $$\int_{a}^{b}L(x)\ ...
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  • 5,287
2 votes

$f(x)>0, f''(x)>0$. Prove: $\int_a^b f(x) dx > (b-a) f(\frac{a+b}{2})$

That is just the Hermite-Hadamard inequality. Up to rescaling we may assume $[a,b]=[-1,1]$ and consider the following: $f(x)$ and $f(-x)$ are strictly convex functions, so $f(x)+f(-x)=g(x)$ is a ...
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$f(x)>0, f''(x)>0$. Prove: $\int_a^b f(x) dx > (b-a) f(\frac{a+b}{2})$

You have the Taylor expansion around the correct point, you just need to only go to the second order term instead of the third order term. That is use that $$f(x) = f\Big(\frac{a+b}{2}\Big) + f'\Big(\...
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How to prove that if $p(x)/||x||^k\to 0$ as $x\to 0$, for a multivariable polynomial $p$ of degree $k$, then $p=0$.

I was actually typing this proof out for myself, so I thought why not share it here. We want to prove the following claim. If $p(x)$ is any $k^\text{th}$ order polynomial in $n$ variables $x=(x^1,\...
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  • 1,375
6 votes
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Is the smooth mapping space a deformation-retract of the continuous one?

No. For example if $X=Y=\mathbb{R}$ then the space of all smooth functions (even polynomials) is dense in $C(X,Y)$. And so it cannot be a retract of $C(X,Y)$. This idea generalizes to any smooth ...
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3 votes
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Is there a handy integrable test function with a polynomial decay and a compactly supported Fourier transform?

For even values of $k$ you may just consider $$\varphi_{2m}(x) =\frac{\cos(x)}{\prod_{n=1}^{m}\left(1-\frac{4x^2}{(2n-1)^2\pi^2}\right)}=\prod_{n > m}\left(1-\frac{4x^2}{(2n-1)^2\pi^2}\right)$$ ...
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What's the mean of " H1(s) be the least element in the set"?

By definition, $H_1(s)$ is the smallest number $n\in \mathbb{N}$ such that $H(n)=s$ (there is always at least one since $H$ is surjective).
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  • 21.2k
1 vote

Let $a>0$. Prove the improper integral $\int_0^{\infty} \cos\left\{{a\over2}\left(x+{1\over x}\right)\right\}x^{k-2}\ dx$ converges for $k>2$

The (improper) integral converges if and only if $0<k<2$. Indeed, considering $a>0$ fixed, for $\lambda\in\mathbb{R}$ and $x\geqslant 1$, put \begin{align*} C(x)&:=\cos\frac{a}2\left(x+\...
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4 votes
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Prove the uniform convergence for the sequence of functions $f_{n}(x)=\sum_{k=1}^{n}\frac{k}{k^2x^2+2n^2}$.

Let $$F(x,t)=\frac t{x^2t^2+2},\qquad x\in\mathbb R, \ t\in[0,1].$$ Then $f_n(x)=\frac1n\sum_{k=1}^nF\left(x,\frac kn\right)$ and thus $$\lim_{n\to\infty}f_n(x)=\int_0^1F(x,t)\,dt=:f(x).$$ We claim ...
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1 vote

Interior Points Question of S in $\mathbb{R}^2$

Yes, for $\mathbb{R}$, every element belonging to $S$ is interior. Yes, $S$ doesn't contain its boundary points. For $\mathbb{R}^2$ case, for any x$(x_1, x_2)$ let $\epsilon = x_1 + x_2$, then ...
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1 vote
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Interior Points Question of S in $\mathbb{R}^2$

From the picture it is 'clear' that $S$ is already open. Assume that $y_1 + y_2 = \epsilon > 0$ Now take $\frac{\epsilon}{2}$ ball around $(y_1, y_2)$. This works since, $|x_1 - y_1| + |x_2 - y_2| &...
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2 votes
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Proving $\lim_{x\to \infty}(a_n + b_n)=\lim_{x\to \infty}(a_n) + \lim_{x\to \infty}(b_n)$ for convergent sequences $a_n$ and $b_n$

I'm not sure what level of rigour you are going for, but here is a pretty standard proof that the limit of sums is equal to the sum of limits (provided everything converges). Assume that $$\lim_{n\to\...
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4 votes

Find the maximum of $\frac{abc}{(4a+1)(9a+b)(4b+c)(9c+1)}$,where$a,b,c>0$

Let $Q = \dfrac{abc}{(4a+1)(9a+b)(4b+c)(9c+1)}= \dfrac{abc}{(b+9a)(4a+1)(4b+c)(9c+1)}$. Apply Cauchy-Schwarz inequality twice: $Q \le \dfrac{abc}{(2\sqrt{a}\cdot \sqrt{b}+1\cdot 3\sqrt{a})^2\cdot(2\...
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4 votes

Find the maximum of $\frac{abc}{(4a+1)(9a+b)(4b+c)(9c+1)}$,where$a,b,c>0$

Hint Try instead to minimize $$\Phi=\frac{(4a+1)(9a+b)(4b+c)(9c+1)}{abc}$$ Compute the partial derivatives All of them being equal to $0$, the solution is immediate.
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1 vote

Why does $1+2+3+\cdots = -\frac{1}{12}$?

Most of the answers here focus either on alternative methods for computing $\zeta(-1)$, or on why the statement "$\sum_{n=1}^{\infty} n = -1/12$" doesn't make sense. While these answers are ...
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1 vote

Lipschitz continuity and and initial condition problem

Let $\tau=\min\{t \ge t_0: x(t) \le \alpha/2\}$ if this closed set is nonempty. (If the set is empty, we are done.) If this set was indeed nonempty, then the mean value theorem would yield some $s \in ...
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1 vote
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How does the Tikhonov regularization work exactly?

Using integration by part on $u'(x)^2 = u'(x)u'(x)$, we obtain, \begin{equation} \int_0^1 u'(x)^2\mathrm{d}x = \left[u(x)u'(x)\right]_0^1 - \int_0^1 u(x)u''(x)\mathrm{d}x. \end{equation} Using the ...
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3 votes
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Question on gradients of the sphere

This is a question about why a certain definition seems useful, so I will provide an example that hopefully illumantes, why it might not be as useful to define $\nabla_{S^{n-1}}f = \nabla f$. In fact, ...
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Calculate area of the graph of the function $f(x,y) = xy$

As you noticed, the graph of a function $f : \mathbb R^2 \to \mathbb R$ is a 3 - dimensional space, a surface. Then, if you are able to parametrice the surface, you can just apply the definition of ...
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3 votes

Is there an accurate way to estimate the product $\prod_{n=1}^{1009}\frac{2n+1}{2n}$?

In the same spirit as @Gary, Stirling approximation gives the series $$\prod\limits_{n = 1}^k {\frac{{2n + 1}}{{2n}}} = \frac{2}{{\sqrt \pi }}\frac{ \Gamma \left(k+\frac{3}{2}\right)}{ \Gamma (k+1)}\...
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4 votes

Is there an accurate way to estimate the product $\prod_{n=1}^{1009}\frac{2n+1}{2n}$?

Let $P(m)=\prod_{n=1}^m{\frac{2n+1}{2n}}$. Using the Wallis product, $$\lim_{m\to\infty}\frac{2m+1}{(P(m))^2}=\frac{\pi}{2}$$ $$\therefore P(1009)\approx\sqrt{\frac{4(1009)+2}{\pi}}\approx 35.9$$
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Bijective function from $[a, b)$ to $(a, b)$

This is not hard using any set theoretical machinery, but interesting in itself anyway. Pick a sequence of numbers $a=a_1<a_2<a_3<\cdots <b$ and $\lim a_n=b$, we can chop $[a, b)$ into a ...
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7 votes

Is there an accurate way to estimate the product $\prod_{n=1}^{1009}\frac{2n+1}{2n}$?

You can use the asymptotics for the ratio of two gamma functions: $$ \prod\limits_{n = 1}^N {\frac{{2n + 1}}{{2n}}} = \frac{2}{{\sqrt \pi }}\frac{{\Gamma (N + 3/2)}}{{\Gamma (N + 1)}} \sim 2\sqrt {\...
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3 votes

Is there an accurate way to estimate the product $\prod_{n=1}^{1009}\frac{2n+1}{2n}$?

The product you are trying to evaluate does not match the product given in the original problem, because the upper index is inconsistent between the two. Specifically, you wrote $$\prod_{n=1}^{\color{...
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5 votes

Is there an accurate way to estimate the product $\prod_{n=1}^{1009}\frac{2n+1}{2n}$?

For the original question with the upper limit of the product $2018$ I would write $$\prod_{n=1}^{2018}\frac{2n+1}{2n}=\frac {4037!!}{4036!!}=\frac {4037!}{(4036!!)^2}=\frac {4037!}{2^{4036}(2018!)^2}$...
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Analyzing the proof of Kirszbraun's theorem

The sets in $(1)$ are closed balls in $\mathbb{R}^n$, so they are compact because they are closed and bounded. The reason why $K_{\gamma}$ is non-empty for large enough $\gamma$ is because for any ...
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-1 votes

Bijective function from $[a, b)$ to $(a, b)$

To do this requires that you know some topology of the real numbers on $(0,1)$. In particular, you need to know about the enumeration of the rational numbers in this particular open interval. Thus let ...
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  • 3,319
2 votes

Calculate the area of the graph of the function $f(x,y) = xy$

We start by calculating the area-element $dA$, which for graphs of functions are: $$dA=\sqrt{1+f_x^2+f_y^2}dxdy=\sqrt{1+x^2+y^2}dxdy$$ Thus the area you are looking for is: $$A = \int_D dA = \int_D\...
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prove the equivalence of sequential and open set def of lower hemicontinuity of correspondence

I know this notion under the name lower semicontinuity. Fact Let $X,Y$ be two metric spaces, $\Phi\colon X\to 2^Y\setminus\{\emptyset\} $ be a multivalued mapping and let $x\in X$ The following two ...
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-1 votes

Calculate the area of the graph of the function $f(x,y) = xy$

Note that $f(-x,y)=-f(x,y)$ and since the unit circle is symmetric with respect to reflection over the y-axis, the area is 0
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  • 1,661
0 votes

Calculate the area of the graph of the function $f(x,y) = xy$

Let's do the change of coordinates (do not forget the jacobian): $$x=r\cos\theta\\ y=r\sin\theta$$ As our area is the unit circle then $r\in[0,1]$ and $\theta\in[0,2\pi]$. So $$A=\int\int_{D}xy\;dxdy=\...
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2 votes

Determine if the following autonomous differential equation has periodic solutions

Hint. $$ \cases{ x x' = -x y\\ y y' = x y + y^3 } $$ after addition $$ \frac 12(x^2+y^2)'= \frac 14 (y^4)' $$
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1 vote

Compactness, open covers and intersection

Let $A,K$ be separated subsets of a metric space. That is, $A\cap\overline K=\emptyset=\overline A \cap K.$ Then $A,K$ are completely separated. That is, there are open sets $U,V$ with $A\subset U$ ...
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2 votes
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Show that the function $f$ in riemann integrable

Let $\varepsilon >0$. For all $p\in\mathbb N$, set $$i_p:=\sup\left\{k\in \{0,...,p\}\mid\frac{k}{p}<\varepsilon \right\},$$ and $$M_i^p:=\sup_{[\frac{i}{p},\frac{i+1}{p}]}f\quad \text{and}\quad ...
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1 vote

Bounds on $\sum\limits_{k=1}^n \frac{\sin(k)}{k}$

This is a summary of my comments to Gabriel Romon's answer, which gave $\displaystyle\left|\sum_{k=1}^n \frac{\sin k}k - \frac12(\pi-1)\right| = \left|\sum_{k=n+1}^\infty \frac{\sin k}k \right|$ $= \...
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3 votes

How to prove the inequation $\int_{c}^{d}f(x)dx ≤\int_{a}^{b}f(x)dx$?

If $[c,d]\subset [a,b]$ then $a\leq c<d\leq b$. Then, we can split the main integral like this:$$\int \limits _a^bf(x)\,dx=\int \limits _a^cf(x)\,dx+\int \limits _c^df(x)\,dx+\int \limits _d^bf(x)\,...
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