Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, and integration through the fundamental theorem of calculus.

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44 views

Proving that an infinite dimensional space is closed.

Let $\mathcal H$ be the subspace of $C([0,1])$ of functions satisfying $f(1-x) = f(x)$ for any $x\in[0,1]$ (these are called even function on $[0,1]$). Then $\mathcal H$ is an infinite dimensional ...
3
votes
2answers
48 views

Calculating in closed form another digamma alternating series

Is there any clever way of finish it fastly? $$\sum _{n=1}^{\infty } (-1)^{n+1} \left(\psi ^{(0)}\left(\frac{5}{8}+\frac{3 n}{8}\right)-\psi ^{(0)}\left(\frac{1}{8}+\frac{3 n}{8}\right)\right)$$ ...
0
votes
3answers
57 views

Given $\phi \in C^{1,b}(R)$, find $\phi_n$ countably piecewise affine functions whose derivatives converge to $\phi'$ uniformly where differentiable

Let $\phi \in C^{1}(\mathbb R)$ with bounded derivative. I am trying to build $\phi_n$ a sequence of countably piecewise affine functions, s.t. $\phi_n'$ converges uniformly to $\phi'$ on $N^c$, where ...
0
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1answer
30 views

Problem about a multivariable calculus

Decide for which of the functions $F:\mathbb R^3\to\mathbb R^3$ given below , there exists a function $f:\mathbb R^3 \to \mathbb R$ such that $(\nabla f)(x)=F(x)$. (A) ...
3
votes
1answer
104 views

Calculating $\int_0^{\pi/4} \frac{\cot (x)}{\cot ^2(x)+\sqrt{\cot (x)}} \, dx$

This is not really one of that kind of integrals that Mathematica cannot handle with, but given the case of a contest, how would we like to handle with it? I would like so much to know your ideas ...
2
votes
1answer
39 views

Fundamental solution Laplace-Poisson equation

Let $\Phi:\mathbb{R}^n\setminus\{0\}\rightarrow\mathbb{R}$ be the fundamental solution of the Laplace equation (see e.g. in the book of Evans). For a function $f\in\mathcal{C}_c^2(\mathbb{R}^n)$ we ...
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0answers
18 views

“Asymptotic” $\mathbb{R}$-algebras

Definition. By an asymptotic $\mathbb{R}$-algebra, I mean an $\mathbb{R}$-algebra $F$ of functions $\mathbb{R} \rightarrow \mathbb{R}$ satisfying: $$\mathop{\forall}_{f:F}\left[\left(\lim_{x ...
2
votes
2answers
41 views

“Scalar product” of two Lp spaces

I was reading the book A. Lasotta and M. C. Mackey, "Chaos, Fractals, and Noise: Stochastic Aspects of Dynamic", Springer, 1991 On page 27, they defined a ``scalar product'' as follows. Let ...
2
votes
1answer
39 views

Picard theorem for ODE.

Theorem : Let $I=]\alpha,\beta[$, $\Omega\subset \mathbb R^n$ an open connexe, $(t_0,x_0)\in I\times \Omega$ and $f:I\times \Omega\longrightarrow \mathbb R^n$, $f=f(t,x)$ such that: $f\in\mathcal ...
1
vote
1answer
67 views

Closed form of this sum

$$\sum _{ s=1 }^{ \infty }{ \left( \frac { 1 }{ 4s-1 } \sum _{ n=0 }^{ \infty }{ \left( \frac { 1 }{ n+1 } \sum _{ k=0 }^{ n }{ \left( \left( \begin{matrix} n \\ k \end{matrix} \right) \frac { { ...
1
vote
1answer
24 views

$\lim_{n\to\infty} d^{-n}e^{o(n)}, d>1$

I think it's rather a silly question, but I have some problems to say definitely what $$ \lim_{n\to\infty}d^{-n}e^{o(n)} $$ is, where $d>1$. Of course, $d^{-n}\to 0$ as $n\to\infty$. Is it true ...
1
vote
3answers
48 views

Equivalence of norms problem.

How would I show that $\|\cdot\|_3$ and $\|\cdot\|_\infty$ are equivalent norms on $\mathbb R^2$? I understand that to say two norms are equivalent, then there exist two real constants, $m,M$ such ...
0
votes
1answer
37 views

Check whether this is indeed a counterexample

Let $A,B \subset \mathbb{R}$; let $Q := A \times B$; and let $f: Q \to \mathbb{R}$ be bounded. The problem is to give a counterexample to the proposition that if the Riemann integral $\int_{Q}f$ ...
-1
votes
1answer
55 views

Are all derivatives of sinc function bounded on real axis?

It seems that all derivatives of $sinc$ function ($sinc(x)=sin(x)/x$) are bounded on real axis. Is it true or no?
-1
votes
1answer
62 views

How to prove the uniqueness of probability measure

Probability essentials P-21 Theorem 4.1 (b) Let $(p_\omega)_{\omega \in \Omega}$ be a family of real numbers indexed by the finite or countable set $\Omega$. Then there exists a unique probability ...
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0answers
13 views

Equidistributed sequences and improper integrals

I could not write mathematic simbols here, so I attach an image for my question:
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1answer
116 views

Is $\overline{D}_{\varepsilon}$ a connected Jordan region in $\mathbb{R}^{n}?$

Definition. Let $E$ be a nonempty subset of $\mathbb{R}^{n}$.The distance from a point $\mathbb{x}\in\mathbb{R}^{n}$ to set $E$ is defined by ...
0
votes
0answers
22 views

From inequality on derivatives to inequality on functions

What is the set of differentiable functions $f$ that satisfy the following inequalities for all $x\geq 0$: $0\leq f'(x)\leq e^{-x}$ Initially, I thought I should just integrate the inequality and ...
-1
votes
0answers
10 views

a,b and c are on complex plane are on the unit circle [closed]

Suppose that a, b and c are on the unit circle in the complex plane and a + b + c = 0. Prove that a, b and c are the vertices of an equilateral triangle. Find the general expression ...
3
votes
0answers
53 views

Check a proof that, besides $\varnothing$, no open set in $\mathbb{R}^{n}$ has measure zero in $\mathbb{R}^{n}$

I am teaching myself Munkres's Analysis on Manifolds and came across an exercise, stated in the title of this question. Please see my proof below and, if doable, criticize it. That a set has measure ...
1
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1answer
39 views

Question about formula for total variation of complex measure from Real Analysis of Folland

Let $\nu$ be a complex measure on $(X, \mathcal{M})$. If $E \in \mathcal{M}$, define: $\mu_1(E) = \sup\{\sum_1^n{|v(E_j)|}:n \in N, E_1, ..., E_n$ disjoint$, E = \bigcup_1^n{E_j}\}$ ...
-1
votes
1answer
25 views

Proving equivalence relations help [closed]

I need to show the three properties of an equivalence relation basically. (a.) Why is A~A for every set A? For this part I know that there exists a function f:A->A that is one to one and onto, ...
0
votes
1answer
52 views

Total boundedness for a non standard metric on $\mathbb{R}^n$

I want to prove this two things: 1) $(\mathbb{R},d_B)$ is not totally bounded. where $d_B=\frac{|x-y|}{1+|x-y|}$ and $d_E$ is the Euclidean metric. 2) $B_M(0)$ is totally bounded in ...
1
vote
1answer
85 views

Does $x_n\sim y_n\implies \sum_{n\geq 1}x_n\sim\sum_{n\geq 1}y_n$

Suppose that $x_n,y_n\geq0$ for all $n$. I know that if $x_n\sim y_n$ and that $\sum_{n\geq 1}y_n$ converge, then $\sum_{n\geq 1}x_n$ converge. But to me, it doesn't imply that $\sum_{k\geq ...
0
votes
1answer
29 views

Transport equations with constant coefficients

Let $X$ be the vector field given by $X = b \cdot \nabla_x + \partial_t$ where $b \in \mathbb{R}^n$ is fixed. Let $f \in C^1(\mathbb{R}^{n+1})$. Assume that $u \in C^1(\mathbb{R}^n \times ...
-2
votes
1answer
62 views

I need help to solve this function [closed]

given that $f(x,y,z)=xy^2-y^2+z^2$ solve $$ \frac{\partial}{\partial x} \left( \frac{\partial f(x,y,z)}{\partial x}+\frac{\partial f(x,y,z)}{\partial y}\frac{\partial y(x,z)}{\partial x}\right)=0 $$ ...
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0answers
30 views

Uniformly boundedness of convolutions

Assume $X$ is an absolutely continuous random variable with pdf $f:\mathbb{R}\to[0,\infty)$. Assume further there exists $M>0$ s.t. $|f(t)|\leq M \quad\forall t\in\mathbb{R}$. Let $X_1,\dots,X_n$ ...
1
vote
1answer
30 views

If partial derivatives w.r.t. x and y are equal at each point (x,y) then which options are correct?

Let, $f$ be a function on $\mathbb R^2$ such that $\frac{\partial f}{\partial x}(x,y)=\frac{\partial f}{\partial y}(x,y)$ for all $(x,y)\in \mathbb R^2$. Then which is(/are) correct? ...
1
vote
2answers
37 views

Suppose $f$ is continuously differentiable on $[0,1]$, $|f'(x)|$ is bounded, show that $|\int_0^1 f(x) - \sum^n_{i=1}f(i/n)\cdot 1/n| \le M/n$.

Problem statement: Suppose $f$ is continuously differentiable on $[0,1]$, and that $\sup_{x \in [0,1]}|f'(x)|\le M< \infty$. Show that $$\big|\int_0^1 f(x) - \sum^n_{i=1}f(i/n)\cdot 1/n\big| \le ...
0
votes
1answer
30 views

A function that satisfies the Intermediate Value Theorem and takes each value only finitely many times is continuous.

I'm having a confusion over the veracity of the statement that a function that satisfies the Intermediate Value Theorem and takes each value only finitely many times is continuous. I've seen from a ...
0
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0answers
38 views

sequence and their arithmetic means

Can it happen that $s_n>0$ and that $\limsup s_n=\infty$, although $\lim \sigma_n=0$ where $\sigma_n=\dfrac{s_0+s_1+\dots+s_n}{n+1}$. I want to take such sequence: $s_n$ is $\frac{1}{n}$ if $n$ is ...
0
votes
1answer
22 views

Find a surjective closed mapping that not an open mapping and a surjective open mapping that is not a closed mapping

Find a surjective closed mapping that not an open mapping and a surjective open mapping that is not a closed mapping Attempt: Suppose $X$ and $Y$ are metric spaces and $f : X \rightarrow Y$. We call ...
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0answers
17 views

Absolute Value Inequality of Differences

I'm hoping someone could give insight as to how I can improve my organization, and/or thought process. Show that $|a-b| \lt c$ if and only if $b -c \lt a \lt b + c$. By the statement $b - c \lt a ...
3
votes
1answer
59 views

Find: $\lim_{n \to \infty} \int_0^{\infty} \arctan(nx) e^{- x^n}dx$

Find: $$\lim_{n \to \infty} \int_0^{\infty} \arctan(nx) e^{- x^n}dx$$ Probably, no recursive form could be found, and elementary tools (integration by parts, change of variable, etc.) are ...
0
votes
2answers
53 views

Show that the mapping $f → f~'$ from $C^1([0 , 1])$ to $C([0 , 1])$ is not continuous.

Let $C^1([0 , 1])$ be the subspace of $C([0 , 1])$ consisting of the functions that have a continuous derivative throughout $[0 , 1]$. Show that the mapping $\Psi:f → f~'$ from $C^1([0 , 1])$ to $C([0 ...
1
vote
1answer
39 views

$f\in BV[a,b]$ has the intermediate value property , then is it true that $f$ is continuous on $[a,b]$ ?

If $f\in BV[a,b]$ has the intermediate value property , then is it true that $f$ is continuous on $[a,b]$ ? Please help . Thanks in advacne
2
votes
1answer
36 views

$f:[a,b]\to \mathbb R$ is continuous , has a finite number of local maxima and minima ; then how to prove that $f$ is bounded variation on $[a,b]$ ?

If $f:[a,b]\to \mathbb R$ is a continuous function having finite number of local maxima and minima ; then how to prove that $f$ is bounded variation on $[a,b]$ ?
0
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0answers
32 views

Absolute Value Inequality Proof

I realize this is almost identical to another question I posted, but I wanted to ask what the distinction between the two is -- comprehension-wise (other than the $\lt$ vs. $\le$). Show that $|b| ...
5
votes
1answer
95 views

How Euler get $\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}$.

I saw on wikipedia that he consider $$\frac{\sin(x)}{x}=1-\frac{x^2}{3!}+\frac{x^4}{5!}+...$$ The roots are given by $x=\pm n\pi$ and thus (to me) $$\frac{\sin ...
2
votes
1answer
29 views

Error estimation for the Wallis product

From the Wallis product we know $$\prod_{k=1}^{\infty} \left(\frac{2k}{2k-1} \cdot \frac{2k}{2k+1}\right) = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot ...
0
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0answers
19 views

Does this theorem concerning upper and lower bnound of a monotone decreasing function have a formal name?

This is the theorem: Let $g$ be a monotone decreasing function and let $a,b \in \mathbb{N}$. Then the following holds true: $$\int_{a}^{b+1}g(x)dx \overset{(i)}{\leq} ...
10
votes
3answers
167 views

Calculating $\int_0^{\infty } \left(\text{Li}_2\left(-\frac{1}{x^2}\right)\right)^2 \, dx$

Do you see any fast way of calculating this one? $$\int_0^{\infty } \left(\text{Li}_2\left(-\frac{1}{x^2}\right)\right)^2 \, dx$$ Numerically, it's about $$\approx ...
4
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0answers
62 views

A discontinuous function $f: X \rightarrow Y$ satisfying: for each closed ball $B$ of $Y, f^{-1}(B)$ is closed in $X$

Find a function $f: X \rightarrow Y$ between metric spaces $X$ and $Y$ that is not continuous but has the property that for each closed ball $B$ of $Y, f^{-1}(B)$ is closed in $X$ Solution Attempt: ...
0
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3answers
44 views

Question about limit of an integral of a continuous function

Let $ f$ be a real-valued continuous function defined for all $0\leqslant x\leqslant 1$, such that $f(0) = 1$, $f(\frac{1}{2}) = 2$ and $f(1) = 3$. Show that $$\lim_{n\to\infty}\int_0^1 \! f(x^n) ...
1
vote
1answer
28 views

What can we say about open unit balls of sup-norm and integral-norm

Consider the normed linear spaces $X_1=(C[0,1], ||.||_1)$ and $X_{\infty}=(C[0,1],||.||_{\infty})$ , where $C[0,1]$ denotes the vector space of all continuous real valued functions on $[0,1]$ and ...
1
vote
1answer
25 views

$P$ is a monic polynomial of degree $n$ , then which are correct?

Suppose that $P$ is a monic polynomial of degree $n$ in one variable with real coefficients and $K$ is a real number. Then which of the following statements are necessarily correct ? If $n$ ...
0
votes
1answer
43 views

How Can I Proof $1+2\cos(\theta)+2\cos(2\theta)+2\cos(3\theta)+.. $ [duplicate]

Proof $1+2\cos(\theta)+2\cos(2\theta)+2\cos(3\theta)+...+2\cos((n-1)\theta)=\frac{2\sin((n-\frac{1}{2})\theta)}{2\sin(\frac{1}{2}\theta)} $
1
vote
1answer
20 views

Finding a differentiable function from (0,1) to (0,1) dominating (point wise) a given continuous function from (0,1) to (0,1)

Suppose we have a continuous function $f: (0,1) \to (0,1)$. Does there exist a differentiable function $\phi: (0,1) \to (0,1)$ such that $f(x) \leq \phi(x)$? There does exist such a differentiable ...
0
votes
2answers
77 views

Example of a set $S$ that is countable, but the set of limit points is uncountable [closed]

What would be an example of a set $S$ so that $S$ is countable. However $S'$ is uncountable. In this $S'$ is the set of all the limit points of $S$.
7
votes
4answers
124 views

If $\int_0^{x/3} f(t)dt =\int_0^xf(t)dt$, prove $f$ is identically $0$

$f:[0,1] \to \mathbf R$ is continuous. If $$\int_0^{x/3} f(t)dt =\int_0^xf(t)dt$$ for all $x$ in $[0,1]$, prove that $f$ is identically $0$. My thought is to prove that the maximum and minimum of ...