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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2
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0answers
120 views

Definition of a bounded complex function and how to apply Liouville's theorem?

The definition of a bounded function is: $$\exists M\in\mathbb{R} \quad st \quad |f(x)| \leq M \quad\forall x\in Domain $$ So consider the complex entire function $f(z)$ such that $Re(f(z))<0$ ...
0
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1answer
42 views

Approximating BV Function by Piecewise Constant Functions

Let $[a,b] \subset \mathbb{R}$ be a bounded interval. A function $f: > [a,b] \to \mathbb{R}$ is of bounded variation if $\sum_{i = > 1}^{n}|f(x_i) - f(x_{i+1})| \leq C$ for all partitions $\{...
1
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1answer
20 views

Why is the derivative of the bounded variation nonnegative definite?

The following is from a proof I am reading. Let $C=((c_{ij}))$ be a continuous, symmetric, $d\times d$ matrix-valued function, defined on $[0,\infty)$, satisfying $C(0)=0$ and $$\sum(c_{ij}(t)-c_{ij}(...
0
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1answer
24 views

On the total variation of a differentiable function

According to Wikipedia the total variation of a differentiable function defined on a bounded open set $\Omega \subset \mathbb{R}^n$ can be expressed as $$V(f, \Omega) = \int_\Omega \left| \, \nabla \, ...
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0answers
25 views

Guess quadratic variation of 2 processes, with same/different BM.

I have 2 processes with stochastic parts R, S. $$dR = \mu_1 dt + \sigma_1 dW_{t1}$$ $$dS = \mu_2 dt + \sigma_2 dW_{t2}$$ I am trying to show what precisely quadratic variation $[R,S]$ means for 2 ...
0
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0answers
43 views

Total variation function

For a function $f : [a,b] \times [c,d] \rightarrow R$, we define $v_f : [a,b] \times [c,d] \rightarrow R$ as $(x,y) \rightarrow V_{f_{[a,x] \times [c,y]}}$. Here $V_f$ is the total variation of $f$ ...
1
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1answer
47 views

Limits, Determinants and Inversion of a matrix-valued function

Suppose I have a matrix-valued, continuous function $$A\colon [0,\infty) \to \mathbb R^{n\times n},\qquad h\mapsto A(h).$$ I know that for the limit $h\to 0$ the matrix is invertible: $$\det\left(\...
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0answers
17 views

RCLL (cadlag) But Not Bounded Variation

In probability theory, we often require that functions and/or processes be RCLL (i.e. cadlag) and of bounded variation (usually bounded on any finite interval). I'm having trouble coming up with an ...
1
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2answers
49 views

Prove that a sequence is bounded/unbounded

I'm trying to do a maths problem which requires me to determine whether a sequence is bounded or unbounded and then it wants me to prove my answe. I know that it's bounded but I've no idea how to ...
3
votes
3answers
140 views

The limit of a sequence of uniform bounded variation functions in $L_1$ is almost sure a bounded variation function

Let $\{f_n\} $be a sequence of functions on $[a,b] $ that $\sup V^b_a (f_n) \le C$, if $f_n \rightarrow f $ in $L_1$ ,Prove that $f $ equals to a bounded variation function almost every where. I ...
1
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1answer
39 views

Continuity and Differentiable functions

Let $f$ be continuous and differentiable on the interval $[a, b]$. Assuming $f$ is bounded on the interval $[a, b]$ and $m = \inf\limits_{[a,b]} f(x)$, prove that there exists $d \in [a, b]$ such that ...
0
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0answers
28 views

Under Given conditions, is f absolutely continuous on $[0,1]$?

1)Let $f: [0,1] \to R$ be a continuous function such that $\mid f(x)-f(y) \mid \le \mid \sqrt{x}-\sqrt{y} \mid $ for all $ x,y \in [0,1]$. Then is $f$ absolutely continuous on $[0,1]$? 2) what about ...
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0answers
28 views

Difference of increasing functions differentiable a.e.

I'm working through Royden & Fitzpatrick's Real Analysis, in the beginning of section 6.3 it reads and I quote: "Lebesgue's theorem tells us that a monotone function on an open interval is ...
0
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1answer
38 views

If given conditions are satisfied, then prove that $f$ is absolutely continuous on any interval $[a,b]$

Assume that $ f: R \to R $ is a non-decreasing function with $ \int_R f' dm =1, $ $ \lim_{x \to-\infty} f(x) =0 $ , $ \lim_{x \to\infty}f(x)=1 $. Then Prove that $f$ is absolutely continuous on any ...
0
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2answers
66 views

total variation and monotonicy

I am stuck with this problem : I have a finite variation function $f$ and I have proved that the total variation on the interval $[0,t]$, denoted with $S_t^f$ for $t\in[0,T]$, is increasing. How can I ...
1
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1answer
26 views

Measure of vanishing set for BV function

Suppose $\Omega$ is open in $\mathbb{R}^N$ and $u \in BV(\Omega)$ is such that $u = 0$ on an open subset $E \subseteq \Omega$. If $D_i u$ denotes the $i$-th partial derivative measure associated to $u$...
2
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2answers
94 views

Unbounded function of bounded variation (in $\mathbb R^d$, $d>1$)

I have seen this stuff tons of times, but every time I see it I got stuck. Today is the right day to clean my ideas once and for all. It is very well known that if $f \colon [a,b] \to \mathbb R$ is ...
1
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1answer
22 views

The upper bound of $L^2$ norm of the minimizer in an minimizing problem.

I am considering the following minimizing problem: $$ u_m:= \operatorname{argmin}_{u\in BV(\Omega)}\{ \frac{1}{2} \|u-u_0\|_{L^2}^2 + t |u|_{TV}\} $$ where $u_0\in BV(\Omega)\cap L^\infty(\Omega)$ and ...
0
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0answers
20 views

The normalization of gradient in weak convergence.

Given $\Omega\subset \mathbb R^2$ is open bounded, smooth boundary, $u_n\in BV(\Omega)$ is bounded in $BV$ norm and in addition we have $$0<\inf |u_n|_{TV}\leq \sup |u_n|_{TV}<+\infty$$ where $|\...
0
votes
1answer
24 views

The zero convergence of total variation

Let $\Omega\subset\mathbb R^2$ be open bounded, smooth boundary. Given a sequence $(u_\epsilon)\subset BV(\Omega)$ such that $$ \|u_\epsilon-u_0\|_{L^2}^2+\epsilon |u_\epsilon|_{TV(\Omega)}\to 0 $$ ...
2
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0answers
58 views

Approximation of Sets of Finite Perimeter

Fix an open set $\Omega \subset \mathbb{R}^n$. If $E$ is a measurable subset of $\Omega$, we may define the perimeter of $E$ in $\Omega$, denoted by $P(E;\Omega)$, to be $$P(E;\Omega) = \sup_{\...
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0answers
34 views

Weak chain rule the other way round.

If $f: \mathbb{R} \to \mathbb{R}$ is weakly differentiable and $u: \Omega \to \mathbb{R}$ is smooth, where $\Omega \subset \mathbb{R}^n$, can we show that for any test function $\varphi \in C_c^1(\...
1
vote
2answers
59 views

Given the function $ f(x) = \sin^2(\pi x) $, show that $ f \in BV[0, 1] $

I'm learning about functions of bounded variation and need to verify my work to this problem since my textbook does not provide any solution : Given the function $ f(x) = \sin^2(\pi x) $, show ...
1
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2answers
73 views

Show that $ f \in BV[0, \pi] $ and find $ V_0^\pi f $ where $ f(x) = \cos^2(x) - 1, \;x \in [0,\pi] $

I'm learning about functions of bounded variation and need some help with this problem : Show that the function $ f $ is of bounded variation on $ [0,\pi] $ and find it's total variation. $$ f(x) =...
0
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1answer
166 views

If a function is not continuous then is it possible for bounded function in given range? [duplicate]

I posted this problem before this .I have satisfied explaination given by markus-scheuer sir and siminore sir . I also found here . I have read the Wikipedia posts for continuous function and bounded ...
1
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1answer
46 views

If $f,f',f''$ are bounded a.e., is $f'$ of bounded variation everywhere?

Assume the function $f$ is such that everywhere except in $0$: $f$ is bounded on $\mathbb{R}$ $f$ is twice differentiable everywhere except in $0$ $f'$ and $f''$ are bounded everywhere except in $0$ ...
0
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0answers
24 views

Bounded variations on $U-\{0\}$ $\Longrightarrow$ bounded variations on $U$?

Given $U\in\mathbb{R}$ a non-empty neighbourhood of $0$ and $f$ a function of bounded variations (BV) on $U-\{0\}$. Can I deduce that $f$ is BV on $U$?
0
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0answers
19 views

The slicing argument for jump set

Given $\Omega\subset \mathbb R^N$ be open bounded, smooth boundary. Let $\nu\in \mathbb S^{N-1}$ be a fixed direction. We define $$ \pi_\nu = \{x\in\mathbb R^N:<x,\nu>=0\},\,\Omega_x = \{t\in\...
0
votes
1answer
32 views

Integrable absolutely continuous $f$ with integrable $f'$ has limit $0$ at infinty

Suppose $f$ is an integrable function on $\mathbb R$. Suppose further that $f$ is absolutely continuous on each closed and bounded interval $[a, b]$, and that its derivative $f'$ also is integrable ...
1
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0answers
34 views

Bounding fluctuations on a random variable

I have some discrete random variable $w$ that has values (in decreasing order) $\underline{w}^\downarrow = \left( w_1, w_2, \dots , w_d \right)$ with corresponding probabilities $\underline{x}=(x_1, ...
2
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0answers
75 views

Concerning the ring of all real valued functions of bounded variation on $[a,b]$

Let $B[a,b]$ the ring of all real valued functions of bounded variation on $[a,b]$ . What is the cardinality of $B[a,b]$ ? How does the maximal ideals of $B[a,b]$ look like ? How does the prime ideals ...
0
votes
1answer
12 views

Bounded variation after a diffeomorphism

This might be a standard property of BV functions but I have not heard about it before. Let $I=[0,1]$ and $\phi:I\to \mathbb R$. We say that $\phi$ is of bounded (total) variation if $$ \sup_{ 0=x_0&...
0
votes
1answer
80 views

What's the total variation of a two-variate step function

What's the total variation of a step function of two variables? Like, $$f(x,y) = \begin{cases} 1 \quad &\text{ when } 0<x<a,\ 0<y<b; \\ 0 \quad &\text{ otherwise}\end{cases} $$ ...
2
votes
1answer
55 views

How do I prove that if $g'$ and $fg'$ are integrable then $f$ is integrable along $g$?

Let $g:[a,b]\rightarrow \mathbb{R}$ be a differentiable function of bounded variation such that $g'$ is Riemann-integrable. Let $f:[a,b]\rightarrow \mathbb{R}$ be a bounded function. If $fg'$ is ...
2
votes
1answer
48 views

Necessary and sufficient condition for Integrability along bounded variation

Related: When is it that $\int f d(g+h) \neq \int f dg + \int f dh$? In this context, I write "integration" to mean the Riemann-Stieltjes integeation Let $g:[a,b]\rightarrow \mathbb{R}$ be of ...
1
vote
1answer
50 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
3
votes
1answer
89 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 $[a,b]$ ?
2
votes
1answer
80 views

Changing the values of a function $f:[a,b] \to \mathbb R$ of bounded variation for countably infinitely many points not a dense subset of $[a,b]$

Let $f:[a,b] \to \mathbb R$ be a function of bounded variation. It is known that if we change its values at finitely many points of $[a,b]$, then the changed function still remains of bounded ...
1
vote
1answer
114 views

If the given function is of bounded variation.

Let $f:[0,1]\rightarrow R$ be a function defined by $ f(x)= 0 $ if x is irrational $f(x)$=${1}\over {q^{2}}$ for $x$=${p}\over{ q} $ where $p$ and $q$ are relatively prime $f(0)=0$ $...
0
votes
1answer
29 views

Does there exist a sequence of partitions $\{P_n\}$ of $[a,b]$ such that $\{V_a^b(f,P_n) \}$ is increasing and converges to $V_a^b(f)$ ?

LEt $f:[a,b] \to \mathbb R$ be of bounded variation . Then does there exist a sequence of partitions $\{P_n\}$ of $[a,b]$ such that $\{V_a^b(f,P_n) \}$ is increasing and converges to $V_a^b(f)$ ?
0
votes
1answer
23 views

If $S$ is any linear space which contains all real valued monotone functions on $[a,b]$ , then $S$ contains all functions of bounded variation

Let $[a,b]$ be a closed bounded interval in real line and $S$ be any set containing all real valued monotone functions on $[a,b]$ such that $f,g \in S \implies f+g \in S$ and $f \in S , c \in \mathbb ...
2
votes
1answer
47 views

Space between $L^1$ and $BV$?

I am looking for a function space $X_s$ such that this space has following properties: $X_s$ is a Banach space, and has lower semi-continuous properties with respect to $L^p$ strong convergence. I ...
3
votes
0answers
51 views

The best constant in Poincare-liked inequality in BV space

Let $\Omega\subset \mathbb R^N$ open bounded smooth boundary. Then we could prove that for any $u\in BV(\Omega)$ and $\omega\in \operatorname{ker}\mathcal E $, where $\mathcal E$ denotes the ...
1
vote
1answer
36 views

The intersection of $BV$ space.

Given $\Omega\subset \mathbb R^N$ open bounded smooth boundary. We define $$ TV(u,\alpha):=\sup\left\{\int_\Omega u\operatorname{div}v\,dx,\, v\in C_c^\infty(\Omega;\,\mathbb R^2),\,\|v\|_{L^\infty}\...
0
votes
2answers
52 views

Prove that $x^\frac{3}{2}\sin (\frac{1}{x})$ is a function of bounded variation for $x\in(0,1]$. [closed]

I am studying for a test in measure theory. Please help with the following question: Prove that $x^\frac{3}{2}\sin (\frac{1}{x})$ is a function of bounded variation for $x\in(0,1]$.
0
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2answers
46 views

Prove that $ \frac {1}{f}$ is a function of bounded variation on $[a,b]$.

I am studying for a test in measure theory. Please help with the following question: Let $f:[a,b]\to R$ a continuous function of bounded variation, when $f(x)\ne 0$ for every $x \in [a,b]$. Prove ...
2
votes
1answer
108 views

Is the space of continuous functions with bounded variation separable?

Let $$BVC([0,1]) = \{f:\mathbb [0,1] \to \mathbb R,\, f\in BV([0,1])\cap C([0,1]),\, f(0)=0\},$$ and $$\|f\|_{BVC}=\mathrm{Var}(f).$$ Is $BVC([0,1])$ separable?
3
votes
1answer
264 views

Total variation measure vs. total variation function

Let $a, b \in \mathbb{R}$ with $a < b$ and define the compact interval $I := [a, b]$. Let $g, h : \mathbb{R} \rightarrow \mathbb{R}$ be non-decreasing and right-continuous on $I$ and constant on $(-...
0
votes
1answer
94 views

When the function is continuous, bounded of variations, absolutely continuous?

Let the function $f_a:[0,1] \to \Bbb R$ be defined by $$f_a(x)=\begin{cases} x^a \cdot \cos(\frac{1}{x}) & 0 < x \leq 1 ;\\ 0 & x=0.\end{cases}$$ Find all values $a\ge 0$ such that ...
0
votes
1answer
35 views

Bounded Variation imply L^1

If a function $f:\mathbb{R}\to\mathbb{C}$ is of bounded variation, is it true that $f\in L^1(\mathbb{R})$? We need to ask for $f$ to be continuous?