For questions about functions $f$ defined on an interval $[a,b]$ such that there exists a constant $M>0$, such that if $a=x_0

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2answers
55 views

Which of the following is true about $f(x)$?

If $f(x)=x+\sin x$, then which of the following is true about $f(x)$? $1.f(x)$ is uniformly continuous on $\mathbb{R}$. $2.f(x)$ has bounded variation on $\mathbb{R}$. $3.f(x)$ does not have ...
0
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1answer
15 views

First Order Variation Example

I am in a introductory stochastic calculus class and came across and example from that asks for the first order variation and the quadratic variation of a continuous function. For example: $f(x) = ...
3
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1answer
44 views

The question regrading to density argument in analysis.

I know the density argument in $L^p$ space, in Sobolev spaces, and even in BV are really sweet in many many cases. However, some times author just work on nice functions and comment by "the rest can ...
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1answer
26 views

Prove that the sum of a trigonometric series has bounded variation

Let $u$: $(0,1)\to \mathbb R$ be defined as $$ u(x):=\sum_{n=1}^\infty \frac{1}{2^n} \cos(2^n\pi x)$$ I am trying to prove that $\operatorname{Var}(u)<\infty$ It is clear that ...
0
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1answer
21 views

The Relation between Holder continuous, absolutely continuous, $W^{1,1}$, and $BV$ functions

I am trying to find out the relation between those spaces. Take $I\subset R$ on the real line. $I$ can be unbounded. Then I have: We first assume $I$ is bounded. If $u\in C^{0,\alpha}(I)$, for ...
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0answers
9 views

reference request about sobolev space and BV space

I am studying Sobolev Space and BV space by using Leoni's and Evans & Gariepy's book. I was wondering that where can I find some explicit example and some computational question of those space. A ...
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0answers
16 views

Bounded/Unbounded sets. [Mandelbrot set]

This is the last question from my assignment. For Part a I have: $z_{n+1}=z_n^2+c$ $\Rightarrow c =z_{n+1}-z_n^2$ $\Rightarrow |c|=|z_{n+1}-z_n^2|=|z_{n+1}-z_n^2||-1|=|z_n^2-z_{n+1}|$ ...
0
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1answer
36 views

Big Omega Proof for $5^n = \Omega(6^n)$

I got the Big-O ($\mathcal{O}$) proof but I am having troubles with the Big Omega ($\Omega$) proof. I am trying to prove $6^n$ is not the tightest bound for $5^n$. Is this logic true: $7^n$, $8^n$ ...
0
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1answer
31 views

The approximation of BV functions

From Evans & Gariepy 's book, I learned that generally for any $u\in BV(R^n)$, we can find $u_n\in BV(R^n)\cap C^\infty (R^n)$ such that $$ \lim_{n\to\infty} \|u_n-u\|_{L^1(R^n)} = 0$$ and ...
0
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1answer
29 views

How to show a given path is NOT rectifiable

We've been given a function and asked to find whether it is a rectifiable path or not. Let $\gamma : [0,1]\to \mathbb{C} $ defined as $\gamma (t)= t + \iota t \sin(\frac{1}{t})$ and $\gamma (0)=0$ ...
0
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1answer
24 views

The convergence in Bounded Variation functions

Given a sequence $u_n\in BV(\Omega)$ and $u\in BV(\Omega)$, where $\Omega\in R^n$ is open. We assume $u_n\to u$in $L^1_{loc}(\Omega)$ and we also assume that $$ \lim_{n\to\infty}|D ...
1
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1answer
23 views

About the definition of variation.

If $f$ is a function of bounded variable on $[a,b]$.Then: $$V_a^b(f)=\sup_{\pi}\sum_{i=0}^n|f(x_{i+1})-f(x_i)|$$ where $\pi$ is a partition of $[a,b]$. I wonder if$$V_a^b(f)=\lim_{\|\pi\|\to ...
-1
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0answers
22 views

Video lecture or examples on functions of bounded variation

I want to get a video lecture or many examples on functions of bounded variation. This is because it is very abstract and hard for me such as Randon measure, singular function , Banach indicatrix and ...
0
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1answer
55 views

Proving budget constraint is compact.

Given the prices $p \in \mathbb{R}_{+}^{k}$ and income $y \geq 0$, define the consumer's budget set as the set of feasible consumption bundles: $\beta(p,y) = \{x \in \mathbb{R}_{+}^{k}: ...
1
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1answer
30 views

Total variation as surface area smooth functions of two variables.

I learnt we have different definitions for the total variation for functions of the form $f:\mathbb{R}^2\to\mathbb{R}$ which are in some way analogous to the total variation of functions of one ...
1
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0answers
17 views

Distributional Representation of Perimeter in Chan-Vese

While re-implementing the classic Chan-Vese Algorithm for image segmentation, I stumbled upon the following statement, which I have problems to understand: Let $H: \mathbb{R} \to \mathbb{R}$ be the ...
1
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1answer
33 views

Books with exercises or problems on the space of functions of bounded variation

I am studying BV space (the space of functions of bounded variation) by using Evans & Gariepy's book. However, there are no exercises for BV space, and I have no idea where I can find some. ...
0
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1answer
25 views

Ways to represent functions of bounded variation as the difference of two monotone functions

I learnt (it is called Jordan's representation theorem) that every function $f : [a,b] \to \mathbb R$ with bounded variation could be represented as the difference $f = g - h$ of two monotone ...
0
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0answers
16 views

Why is the weakest bound only possible at these specified mins

Given the following: $$0 < g_0 \leq g_1$$ $$0 \leq B_0 \leq B_1$$ $$D_0 \leq D_1$$ (note: these are the only two variables that could be negative) $$0 \leq c_0$$ $$0 \leq c_1$$ $$0 < f_0$$ ...
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2answers
43 views

Property of the variation of a function

I need help with the following: given $ f:[a,b]\rightarrow \mathbb{R}$, show that $$V_f (a;b)=V_f(a;c)+V_f (c;b)$$ with $a< c <b$. We know that $$V_f(a;b) \geq \sum_i |f(x_i)-f(x_{i-1})|$$ for ...
1
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1answer
78 views

Maximizing a particular integral / functional

I have a (probably simple) question whose answer seems obvious but I cannot prove it. It relates to the calculus of variations. Let scalar $A = \Re[\int_a^bB(x)C(x)dx$], where $B$ and $C$ map ...
1
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0answers
19 views

Change of variables formula with integrator of bounded variation

Let $G$ be be continuous with bounded variation on finite intervals. If $f$ is continuous then it is well known that $\int_a^bf(G(s))dG(s)=\int_{G(a)}^{G(b)}f(x)dx$. How general can $f$ be so that ...
2
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1answer
86 views

Unbounded variation but differentiable everywhere

A function with bounded variation is differentiable almost everywhere. There are also functions with unbounded variation that are differentiable almost everywhere (e.g. take ...
0
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1answer
28 views

Definition of the total variation of a function $g:\mathbb{R}\to\mathbb{R}$

if the total variation of a a real function $f:[a,b]\to \mathbb{R}$ over $\textbf{P}=\{a=t_0<t_1<...<t_m=b\}$ is $$ V^{a}_{b}(f)=\sup_{\textbf{P}}V(f,\textbf{P}) $$ where $$ ...
1
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0answers
22 views

Evaluate the bound of the following complex integral

Let $q$ be a complex analytic function such that $\left|q\right|\neq0$ everywhere in the domain, except at the boundaries ($\left|q\right|\rightarrow0$ as $z\rightarrow\pm\infty$). Is it possible ...
1
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1answer
62 views

A problem involving Stieltjes Integral and bounded variation

I found this problem in a book I'm using to study (Curso de Análise - Vol 2, Elon Lages Lima). "Let $\alpha:[a,b] \to \mathbb{R}$ be a bounded function. If $\displaystyle \int_{a}^{b}f(t)d\alpha$ ...
1
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0answers
38 views

Question on Bounded Variation involving partitions

If $f\colon[a,b]\rightarrow \mathbb{R}$ and $P=(a=x_0, x_1, ...,x_n=b)$ is any partition of $[a,b]$ let $v_f(P)$ be defined as $$v_f(P) = \sum_{k=1}^n |f(x_k) - f(x_{k-1})| $$ and the total variation ...
1
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1answer
40 views

Question on Functions of Bounded Variation

We say that $f$ is of bounded variation over $[a,b]$ or $f\in BV[a,b]$ if $V_f[a,b] = \sup \sum_{k=1}^n |f(x_k) - f(x_{k-1})| < \infty, $ where the supremum is taken over all possible partitions ...
1
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1answer
68 views

Does bounded and continuous implies Lipschitz?

If a function $f : \mathbb{R} \rightarrow \mathbb{R}$ is integrable, bounded and continuous, is it also Lipschitz continuous?
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0answers
46 views

Does the fundamental theorem of calculus hold for BV functions?

I am a bit confused and I hope you can help me in understanding a bit better these things. Let us start by considering one dimensional case. Let $f\colon \mathbb (a,b) \to \mathbb R$ be a function. ...
3
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0answers
42 views

Understanding the duals of $L^{\infty}(K)$ and $C(K)$ and weak-* compactness

Let $(K\subset \mathbb{R}^n,\mathcal{B}(K),\lambda)$ be a measure space, where $\lambda$ is the Lebesgue measure and $K$ is compact. According to Wikipedia (with adapted notation), The dual ...
1
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1answer
37 views

Does bounded variation and continuous means total variation continuous

$F$ is of bounded variation and continuous. Is it true that total variation is continuous ? In case, $F$ is absolutely continuous it is trivial to see. But for the above case how to proceed ?
2
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2answers
40 views

Continuity and Inverse Preservation of Boundedness Implies Preservation of Closed Sets

Sorry about the rather long title - wasn't sure what else to call it! Here is my question: Let $f : \Bbb R^n \rightarrow \Bbb R^m$ be continuous and such that $f^{-1}(F)$ is bounded whenever $F$ ...
1
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1answer
43 views

One inequality involving Total variation function

If $F$ is of bounded variation in $[a,b]$, then I need to prove that $$ \int_{a}^{b}|F'(x)| dx \leq T_F(a,b)$$ If $F'$ were Riemann integrable then it was easy to prove (in fact we can prove ...
1
vote
1answer
52 views

Does bounded variation imply boundedness

Using the standard definition $$||f||_{TV} := \sup_{x_0<\cdots<x_n}\sum_{i=1}^{n} |f(x_i) - f(x_{i-1})|.$$ 1.When the domain is a bounded interval $[a,b]$, the statement holds. 2.When the ...
0
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0answers
38 views

Question about deformation

Why if $H\setminus X$ has no crititical point then $X$ is a deformation retract of $H$? $H$ is a Hilbert space $X$ is a subspace of $H$ ($X=\varphi^b=\lbrace x\in H, \varphi(x)\leq b\rbrace$ , ...
3
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1answer
135 views

Function coincides with a function of bounded variation almost everywhere

Problem Suppose $f\in L^1(\mathbb R)$ satisfies that there exists $A\ge0$ such that $$\int_{\mathbb R}\lvert f(x+h)-f(x)\rvert dx\le A\lvert h\rvert$$ for all $h\in\mathbb R$. We need to show that ...
0
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0answers
31 views

Gaussian independent, Mean, expectation, variance

Let $X$ and $Y$ be two independent Gaussian variables with zero mean and variance $\sigma^2$. Define:$$Z = |X-Y|.$$(a) Show that $\operatorname{E}[Z]= 2 \sigma / \sqrt{\pi}$. (b) Show that ...
0
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0answers
29 views

How to represent?

You are a well-known hedge fund manager in Wall Street circles. One of your wealthy clients has $\$1$ million dollars to invest in XYZ stocks. Currently, XYZ stocks are trading at $\$2$ per share. You ...
0
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0answers
79 views

Upper bound for logarithmic integral

If i'm not mistaken $li(x)=O(x/logx)$ So we can write $li(x)≤ cx/logx$ . For x>1 tell me any c>0.5 ... I really need this...
0
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1answer
29 views

A question related to bounded variation

Let $f\in C^{BV}([0,1])$ (i.e. continuous and has bounded variation). Let the intervals $I$ and $T$ satisfy the following: $I\subset T\subset [0,1]$ and for sufficient small $\delta>0$, ...
3
votes
1answer
71 views

Definition of total variation

According to wikipedia, the total variation of the real-valued function $f$, defined on an interval $[a,b]\subset \mathbb{R}$, is the quantity $$V_b^a=\sup_{P\in\mathcal{P}}\sum_{i=0}^{n_P-1}\left | ...
3
votes
1answer
63 views

Differentiable function in n dimensions

If a function $f:\mathbb R^{n} \rightarrow \mathbb R^m $ is differentiable at a point $a$ can we say that there is a neighbourhood of $a$ such that $f$ is locally Lipschitz? (i.e. there is some ...
0
votes
1answer
70 views

Bounded variation

Define $$g(x) = \begin{cases} x^2\cos\left(\frac{1}{x}\right) &\mbox{if } x \neq 0\\ 0 & \mbox{if } x = 0. \end{cases} $$ Is g of bounded variation on $[-1,1]$. My attempt:-To show that $g$ ...
0
votes
1answer
45 views

Problem on Bounded Variation

Assume $f$ is of bounded variation on $[a,b]$. Show that there is a sequence of partitions $\{P_n\}$ of $[a,b]$ for which the sequence $\{TV(f,P_n)\}$is increasing and converges to $TV(f)$? ...
1
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2answers
43 views

Need guidance on a problem about oscillation from Spivak's Calculus on Manifolds

I've been stuck on this particular problem for a while now: Let $f: [a,b] \rightarrow \mathbb{R} $ be an increasing function. If $x_1, ... ,x_n \in [a,b]$ are distinct, show that $\sum\limits_{i=1}^n ...
2
votes
0answers
82 views

Arzela Ascoli variation theorem

I am trying to prove the following variation of the Arzel-Ascoli theorem: Let $(f_n)n$ be a sequence of differentiable functions on $[a,b]$ whose derivatives are uniformly bounded and there is an ...
0
votes
1answer
23 views

Definition of Bounded Variation Function with vectorial arguments

I try to found a definition of a function $$f(x)\colon\mathbb R^m\to \mathbb R^n$$ that use the norm. Is the formula below correct? $$TV=\sup\sum_{i=1}^k \|f(x_i)-f(x_{i-1})\|.$$ with k any finite ...
2
votes
1answer
49 views

Monotone convergence for monotone functions in BV

For $n \geq 1$ let $f, f_n : [0, 1] \to \mathbb{R}$ be monotone nonincreasing functions. Suppose that $f_n \nearrow f$ pointwise monotonically as $n \to \infty$. Is then $\mathrm{TV}(f - f_n) \to 0$ ...
1
vote
1answer
18 views

Trying to look for a lower bound on distance between distributions

I have two distributions $X$ and $Y$, is it true that for any arrangement of $X$ and $Y$ values $d= \sum_{i=1}^{n}(x_j-y_j)^2$ follows $d\geq\sum_{i=1}^{n}(x_i-y_i)^2$ where $x_i$ and $y_i$ are the ...