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, integration through the fundamental theorem of calculus.

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53 views

Looking for a hint on the following integration problem

Let f(x) be continuous on [0,1]. Calculate $\lim_{n \to \infty} n\int_0^1 x^n f(x)dx$. What immediately jumps out at me is how close $\frac{x^nf(x)}{\frac{1}{n}}$ looks to a derivative, i.e. if ...
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0answers
5 views

Remainder of the minimax approximation polynomial - number of extrema

Recall some definitions. Let $f \in C [a,b]$. The minimax polynomial $p_n$ is the polynomial $p_n (x) $ of degree $\leq n$ that minimizes $||f-p_n||_\infty $. It can be proved that this polynomial ...
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17 views

Initial value problem with intermediate value

The Picard Lindelöf theorem I know always assumes that we specify the value at the left end of the time-interval Picard Lindelöf. Is it true that $x'(t) = f(x(t))$ has a unique solution, in an open ...
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2answers
52 views

$\Gamma \subset \mathbb{R}^{+}$ is uncountable. Can we choose a sequence from $\Gamma$ of which the sum is $\infty$

If $\Gamma$ is a set of uncountably many different positive real numbers, can we choose a sequence of pairwise different positive numbers from $\Gamma$, say $\{a_n\}$, such that $\sum a_n = \infty$ ...
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1answer
26 views

Maximum value of function involving factorials

Define $$g_{(k,j)} = \frac{a^{n-k}b^k(k+n)!x^{k+n-j}}{k!(n-k)!(k+n-j)!}$$, where $n,k,j \in \Bbb{N}$ are fixed such that $(0 \leq x \leq a/b ),(b<a),(0 \leq k \leq n ),(2 \leq j \leq 2n),(0 \leq ...
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20 views

gradient of a differentiable real function [on hold]

let f is a differentiable real function in R^n. If f is not 0, then prove $\nabla(1/f)=-(1/f^2)\nabla(f)$ can this be proved by chain rule?
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1answer
32 views

Generalization of Bernoulli's Inequality

Is it possible to generalize Bernoulli's Inequality to $(x+y)^n \geq x + ny$ provided $x+y \geq 0 $ and $x \geq 1$ and $n$ is a positive natural number? I was thinking that the proof follows by ...
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3answers
37 views

Series Convergence/Divergence $\frac{n^n}{(n+1)^{n+1}}$

Trying to establish whether $\sum x_n$ for $x_n := \frac{n^n}{(n+1)^{n+1}}$ converges or diverges. Here's what I've done so far: 1) n-th term: $x_n < \frac{n^n}{n^{n+1}} = \frac{1}{n}$, so ...
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2answers
61 views

Proving Lebesgue integration result

I have a Lebesgue integration question and a proposed proof. Please advise. Let $\Omega \subset \mathbb{R}^{n}$(denote the boundary as $\partial \Omega$) and consider $$\int_{\partial \Omega} vf ...
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35 views

One example of this theorem wanted?

One example of this theorem wanted? Let $ \Lambda =\{\lambda_{n}\}_{n=1}^{\infty} $ be a sequence of distinct complex numbers satisfying the following two properties: (1) $ ...
2
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2answers
19 views

How many tagged partitions of an interval are there?

A tagged partition of an interval $[a, b] \ (a, b ∈ ℝ, a < b)$ is a finite sequence $(x_i)_{i=0}^n$ in $ℝ$, where $a=x_0 < x_1 < … < x_n = b$. Consider the set of all tagged partitions of ...
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1answer
31 views

Characteristic function of Cantor set is Riemann integrable

I want to prove that the characteristic function of the Cantor set is Riemann integrable on $[0,1]$. Could somebody please tell me if my proof is correct? Let $f$ be the characteristic function of ...
2
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3answers
61 views

What is the sufficient condition for the value of integrable function $f\in L^1(\mathbb{R})$ to go to $0$ when $|x|\rightarrow \infty$?

What is the sufficient condition for the value of integrable function $f\in L^1(\mathbb{R})$ to go to $0$ when $|x|\rightarrow \infty$? Case 1: $f $ is differentiable on $\mathbb{R}$. Case 2: $f$ is ...
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31 views

Set of points of non-differentiability

As an exercise I proved that if $f: \mathbb R \to \mathbb R$ is a function then the set of discontinuities of $f$ must be an $F_\sigma$ set. I thought it was an interesting result. Now I am ...
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1answer
32 views

Show that there exist an open set $V$ containing $s=1$

Let $$m=\lim_{s→1}\frac{(s-1)f′(s)}{f(s)}$$ where $f$ is an analytic function with $s=1$ as a zero of order $m$. My question is: By using the definition of the limit show that there exist an open ...
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36 views

p-norm of a function

Let $f\in L^1(\mu)\cap L^\infty(\mu)$. I have proved for any $1<p<\infty$, $f\in L^p(\mu)$, $w(p)=||f||_p$ is continuous w.r.t. $p$, and $\lim_{p\to \infty}||f||_p=||f||_\infty$. Is $w(p)$ ...
2
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1answer
49 views

asymptotical behavior of integral

I'm interest in the asymptotical of $$\int_{-\pi}^{\pi}\exp\Big((\cos z+i\alpha\sin z-1)t\Big)dz\hspace{3mm}\text{as}\hspace{2mm}t\to\infty$$ for $-1<\alpha<1$. Numberical result suggest that ...
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1answer
30 views

Uniform and integral limit

Let $f_n(x)=n(\sin x)^n \cos x$. Show that the sequence of functions $f_n$ converges to $0$ uniformly on any interval of the form $[0,a]$ where $a<\pi /2$. Show that, for any continuous function ...
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30 views

Seeing that a function is a trigonometric polynomial

I'm working through Chapter 4 of Rudin's Real and Complex Analysis book right now, and I've found myself rather more confused than usual. In the proof of the completeness of the trigonometric system, ...
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1answer
53 views

A question on limit of a sequence

Suppose that $\varphi(n)$ is a positive monotone increasing function defined on $N$ and $\lim_{n\to \infty}\frac{\varphi(n)}{n}=0$. Let $\{n_k\}$ be a subsequence with $\lim_{k\to ...
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1answer
33 views

Convergence of measures — revisited

In this thread, I asked a question about the convergence of measures. The conjecture I posed there, which turned out to be false, was supposed to be a lemma that I wanted to use to prove a ...
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5answers
98 views

How I could show that :$\log1=0$?

I would be like somone to show me or give me a prove for this : Why $\ln 1=0$ ? Note that $\ln$ is logarithme népérien, the natural logarithm of a number is its logarithm to the base $e$. Thanks ...
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41 views

a complicated question about double improper integral

how to evaluate $$\iint_{y\ge x^2+1}{dx\,dy\over{x^4+y^2}}$$ My solution: the initial intergral $$ =2\int_0^\infty \left(\int_{x^2+1}^\infty {dy\over {x^4+y^2}}\right)\,dx = \int_0^\infty ...
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31 views

Relation between measurable sets

Given an outer measure $\mu^*$ with domain $X$ and a subset $B$ of $X$, one can construct a new outer measure $\nu^*(A) = \mu^*(A \cap B)$. The problem is to find the relation between the measurable ...
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1answer
38 views

Span of Dirac's delta distributions dense in Hilbert space of $L^2$ functions?

According to Wiki a set of elements of a Hilbert space(B) is a basis for that space if: Orthogonality: Every two different elements of $B$ are orthogonal: $⟨e_k,e_j⟩=0$ for all $k$, $j$ in $B$ with ...
3
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1answer
26 views

Show $p'(x) + kp(x)$ has a zero between consecutive roots of the polynomial $p$

For any $k \in \mathbb{R}$ and any polynomial $p(x)$, show that $$p'(x) + kp(x)$$ has a zero between consecutive roots of $p$. I have tried writing $p(x) = a_n x^n + \cdots + a_0$, but this does not ...
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22 views

differential equation with random coefficient

I am confused with a problem I encountered at hand, not on how to work on it but rather understanding the problem itself: Let $A(x;\omega)$ be a random field taking values in $[a,b]$ where $a,b < ...
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1answer
26 views

On seqeunce of functions $h_n$ satisfying $\Vert\sum_{n=1}^\infty f\ast h_n\Vert_1=\sum_{n=1}^\infty\Vert f \ast h_n\Vert_1$ for all $f\in L_1(G)$

Let $(h_n)$ be a sequence of non-zero functions in $L_1(G)$ (where $G$ is a locally compact group) with the property $$ \left\Vert\sum_{n=1}^\infty f\ast h_n\right\Vert_1=\sum_{n=1}^\infty\Vert f ...
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1answer
36 views

New variable in a convex optimization problem

Consider the convex optimization program $$ \min_{x \in X } x^\top P x + p^\top x \quad \text{ sub. to: } Ax = b $$ where $X \subset \mathbb{R}^n$ is compact, $P \succ 0$, $A \in \mathbb{R}^{m \times ...
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1answer
43 views

Convergence of Taylor series of $\sqrt{1-x}$

Concerning $$\sqrt{1-x} = \sum_{k=0}^{\infty} \left[\prod_{j=1}^k \left(\frac{j-1-\frac{1}{2}}{j}\right)\right]x^k$$ the Taylor series about $x=0$. For $|x|< 1$ this series converges uniformly. ...
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2answers
34 views

For $f, g \in C^1$, $fg' - f'g \neq 0$ implies that the zeros interlace

Let $f, g \in C^1$, and suppose that $f(x) g'(x) - f'(x) g(x) \neq 0$ for all $x$. Show that The roots of $f$ do not have an accumulation point. The roots of $f$ and $g$ interlace, so that if ...
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2answers
36 views

What functions satisfy the condition $f(x,y)=g(x)$?

Are there any functions $f(x,y)$ and $g(x)$ that satisfy (1) $f(x,y)=g(x)$ for all $x \in \mathbb{R}$ and $y\in \mathbb{R}$ (2) $f(x,y)$ is not constant in $y$ for each $x$ (i.e. for each $x$ there ...
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49 views

Integral limit of $\sin(x/n)f(x)$

For any $f\in L^1[0,\pi]$, evaluate $n\to \infty \int^\pi_0 n$sin$(x/n)f(x)dx$ My idea is, $n$sin$(x/n)f(x)\to xf(x)$ and it seems that it is increasing sequence. I am not able to show it is ...
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2answers
22 views

Text on convergence theorems in probability theory (various modes of convergence)

I need a text reviewing theorems and discussing with details ALL the types of convergence in probability theory such as almost sure convergence, convergence in probability, weak convergence, $L^p$ ...
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1answer
44 views

Homeomorphism are equivalence relations, so what are the equivalence classes?

Homeomorphisms are equivalence relations, so what are the equivalence classes for two Topological spaces $T_1, T_2$? Intuitively it seems like we might have the following equivalence classes - ...
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1answer
35 views

What smooth functions are solutions of an autonomous ODE?

Let $y$ be a smooth function, say $y : \mathbb{R} \rightarrow \mathbb{R}$. When can we find a continuous map $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $y'=f(y)$ ? Obviously it's not always ...
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1answer
54 views

$f(A) \cap f(B) = f(A \cap B)$ if $f$ is a bijection?

I found this statement in a Topology proof - $$f(A) \cap f(B) = f(A \cap B)$$ if $f$ is a bijection I haven't come across this statement before. Is this some axiom of set theory?
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1answer
37 views

Intuition behind homeomorphism from $B((0, 0), 1) \to \mathbb{R^2}$

In my notes I have that the following function is a homemorphism from $B((0, 0), 1) \to \mathbb{R^2}$ $$h(x, y) \to \frac{f(\sqrt{x^2 + y^2})}{\sqrt{x^2 + y^2}} (x, y)$$ where $f = ...
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2answers
29 views

Find a function $f$ which is $L^p$ integrable but not $L^\infty$ integrable satisfies $\int_{supp(f)} e^{|f|} dx =1$

Find a function $f$ which is $L^p(\mathbb{R})$ integrable but not $L^\infty(\mathbb{R})$ satisfies $\int_{supp(f)} e^{|f|} dx =1$. I have no idea how to find such function belongs to $L^p\cap ...
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1answer
38 views

is bounded partial derivative continous

Let $f:{\mathbb R}^2\rightarrow {\mathbb R}$ be defined as: $$ f(x,y) = \left\{ \begin{array}{ll} \frac{x^3}{x^2 + y^2}, & \ (x,y)\ne(0,0),\\ 0, & \ (x,y)=(0,0).\\ \end{array} \right. $$ Prove ...
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3answers
38 views

Show $A$ is unbounded given $\int_A f'(x) dx \leq 0$

Let $A\subset \mathbb{R}$ be a non-empty open set such that $$\int_A f'(x) dx \leq 0$$ for all $f\in\mathcal{C}_c^1(\mathbb{R})$ with $f\geq 0$. Prove that $A$ is unbounded. The hint says first show ...
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43 views

Is a continuous function $f$ from a metric space $(X, d)$ to $\mathbb{R}$ compact under some certain condition?

$(X, d)$ is a metric space with metric $d$ and there is $x_0 \in X$ and define $E_\epsilon=\{x\in X, d(x, x_0)\geq \epsilon\}$. If $f$ is continuous function from $X$ to $\mathbb{R}$ such that ...
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42 views

Is $\sin (\mathbb N)$ dense in $[-1,1]$ ? [duplicate]

Let $\mathbb N$ be the set of +ve integers , then is it true that $\sin (\mathbb N)$ is dense in $[-1,1]$ i.e. is it true that for every $x,y \in [-1,1]$ with $x<y$ , $\exists m \in \mathbb N$ such ...
4
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1answer
31 views

Proving this function is differentiable at $1$

Define $h(x) = 1$ except at $1$ where $h(1) = 0$. Also define $H(x) = \int_0^x h(t)$. Now I tried to show that $H$ is differentiable at $1$. My proof is to compute $$ \lim_{x \to 1^-} {H(1) - ...
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22 views

Lipschitz at a point implies Lipschitz in a neighborhood

A function $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is said to be Lipschitz at a point $x \in \mathbb{R}^n$ if there is a neighborhood $N_x$ of $x$ and $L > 0$ such that $$ |f(y) - f(x)| \leq L ...
0
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1answer
21 views

Unique solution for $\int_x^1 f(t) dt = 2x$ and $|x| < \epsilon$

Let $f$ be continuous on $\mathbb{R}$ such that $$f(0) \neq -2 \quad\text{ and } \quad \int_0^1 f(t) = 0.$$ Show that there exists $\epsilon > 0$ such that the equation $$\int_x^1 f(t) dt = 2x$$ ...
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2answers
11 views

A cycloid that goes through the beginning and through a general point

Parametric equations of the general cycloid through the beginning $(0,0)$ are $$x(t)=\frac{2t-\sin2t}{2d}$$ $$y(t)=\frac{1-\cos 2t}{2d}$$ How can we determine $d$ such that the cycloid goes through ...
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2answers
33 views

Convergence of measures

Consider the measurable space $(\mathbb R,\mathscr B_{\mathbb R})$. Suppose that $\{\mu_n\}_{n=1}^{\infty}$ is a sequence of finite Borel measures and $\mu$ is a finite Borel measure on $\mathbb R$. ...
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0answers
29 views

A variant of the Riemann Integral

This question is related to this one. Let $S_k$, $k\in\mathbb{N}$ be a sequence of finite sets where $S_k\subset S_{k+1}\subset[0,1]$. Fixed $s$ in $S_k$, let $s'$ denote the predecessor of $s$ in ...
5
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3answers
65 views

Is the set $\{ x\in \mathbb{Q}: 2< x^2 <3\}$ closed, bounded, compact in $\mathbb{Q}$?

Is the set $\{ x\in \mathbb{Q}: 2< x^2 <3\}$ closed, bounded, compact in $\mathbb{Q}$ ? I think $\{ x\in \mathbb{Q}: 2< x^2 <3\}=\{ x\in \mathbb{Q}: 2\leq x^2 \leq 3\}$, so it is bounded ...