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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11 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 ...
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1answer
22 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. ...
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1answer
21 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 ...
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0answers
15 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
40 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 ...
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1answer
64 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 ...
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0answers
16 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 ...
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1answer
83 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 ...
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1answer
25 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 $$ ...
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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 ...
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1answer
59 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 $\displaystyle \int_{a}^{b}f(t)d\alpha$ ...
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0answers
33 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 ...
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1answer
33 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 ...
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1answer
56 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
38 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. ...
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0answers
38 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 ...
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1answer
35 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 ?
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2answers
33 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 $F$ ...
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1answer
42 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 ...
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1answer
42 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 ...
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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 $H$ ($X=\varphi^b=\lbrace x\in H, \varphi(x)\leq b\rbrace$ , ...
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1answer
132 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 ...
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0answers
28 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 ...
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0answers
28 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 ...
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0answers
77 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...
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1answer
26 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
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1answer
61 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
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1answer
58 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 ...
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1answer
65 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 $g$ ...
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1answer
42 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 $TV(f)$? ...
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2answers
41 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 ...
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0answers
68 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 ...
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1answer
21 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 ...
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1answer
45 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 $\mathrm{TV}(f - f_n) \to 0$ ...
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1answer
17 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 ...
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2answers
35 views

Find limits of value/derivatives defining a polynomial at 2 points to bound it in between

The following properties define the polynomial $p(x)$ uniquely: $\text{deg}(p(x))=7\\p(-1)=y_1,\ p'(-1)=d_{1,1},\ p''(-1)=d_{2,1},\ p'''(-1)=d_{3,1}\\ p(1)=y_2,\ \ \ \ p'(1)=d_{1,2},\ \ \ \ ...
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1answer
27 views

Bounded & Norm space [closed]

Can someone help me on this exercise ? Thanks!
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1answer
149 views

Uniformly convergent subsequence of Equicontinuous R to R functions which are bounded in a point

Suppose that $(f_n)$ is an equicontinuous sequence of functions $f_n : \mathbb{R}\rightarrow \mathbb{R}$, such that $(f_n (0))$ is a bounded sequence in $\mathbb{R}$. Does there exist a subsequence ...
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0answers
17 views

A question on a function of bounded semivariation(part 2)

Let $(X,\tau)$ be a Hausdorff locally convex topological vector space and let $P(X)$ be a family of continuous seminorms on $X$ that generates the topology $\tau$. I got the following definition from ...
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1answer
18 views

Is a function of bounded semivariation bounded?

Let $X$ be a topological vector space and let $f:[a,b]\to X$. We say that $f$ is of bounded semi-variation in $[a,b]$ if the set $SV(f,[a,b])$ consisting of all the elements of the form $$\sum_{i=1}^n ...
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1answer
31 views

Partitions of $\alpha$ Variation

Suppose $T>0$ . Does anyone know if there exists a sequence of partitions $(\pi_n)_{n\in\mathbb{N}}$ of the interval $[0,T]$ such that the mesh size goes to $0$, and such that it is of bounded ...
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1answer
46 views

Total variation for functions, Meaning of supremum as used here?

On Wikipedia article, here: http://en.wikipedia.org/wiki/Total_variation, on definition 1.1 there says, "where the supremum runs over the set of all partitions ..." AFAIK supremum is defined for a ...
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3answers
2k views

If a function $f(x)$ is Riemann integrable on $[a,b]$, is $f(x)$ bounded on $[a,b]$?

Most statements regarding Riemann integrals (at least the ones that I have encountered) begin with the statement "for $f(x)$ bounded on $[a,b]$." I am wondering if Riemann integrability implies ...
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1answer
81 views

Finding the total variation of $3x^2-2x^3$ [closed]

I would appreciate if someone could help me to find the total variation of $3x^2-2x^3$ on $[-2,2]$. Thanks
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1answer
301 views

Is $x\sin(1/x)$ of bounded variation?

I can't figure out whether $f(x)$ where $f(x)=x\sin(1/x)$ $f(0)=0$ is of bounded variation on $[0,1]$ or not. But I think it is not. Can someone suggest a partition to prove it is not of bounded ...
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1answer
60 views

A question on a function of bounded semivariation

Let $(X,\tau)$ be a Hausdorff locally convex TVS and let $P(X)$ be a family of continuous seminorms on $X$ that generates the topology $\tau$. I got the following definition from one of the articles ...
3
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2answers
420 views

If f is of bounded variation is f Riemann integrable?

I want to know if f is of bounded variation on [a,b] does it follow that f is Riemann integrable on [a,b]?
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1answer
19 views

A problem that involves a supremum

We let $[a,b]\subset \mathbb{R}$ and $f:[a,b]\to \mathbb{R}$. How do we prove that $$\sup \left|\sum_{i=1}^n a_i[f(x_i)-f(x_{i-1})]\right|=\sup\sum_{i=1}^n|f(x_i)-f(x_{i-1})|$$ where $a_i\in [-1,1]$ ...
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1answer
38 views

Function of bounded variation and cardinal of set of discontinuities.

Let $f:[a,b] \to \mathbb R$ such that the image is a finite set. Prove that $f$ is of bounded variation iff the set of discontinuities of $f$ is finite. I didn't have problems with the forward ...
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1answer
135 views

Relation of total variation of a function $f$ and the integral of $|f'|$

Let $f:[a,b] \to \mathbb R$ a function of class $C^1$ on the interval $[a,b]$. Prove that: i) $f$ is a function of bounded variation. ii) The equality $V_a^b f= \int_a^b |f'(x)|dx$ holds. My ...