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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27 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 ...
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1answer
37 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 ...
-1
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2answers
63 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
25 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 ...
2
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2answers
89 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
19 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
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1answer
23 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
57 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) = ...
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0answers
33 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 ...
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2answers
57 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) ...
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1answer
164 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 ...
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1answer
45 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 $0$ ...
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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 $U$?
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0answers
18 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 = ...
0
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1answer
29 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
28 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
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1answer
11 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
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1answer
74 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
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1answer
53 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
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1answer
45 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
86 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
77 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
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1answer
110 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
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1answer
27 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
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1answer
22 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
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1answer
46 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
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0answers
45 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
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1answer
34 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 ...
0
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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
104 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
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1answer
244 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
90 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
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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?
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0answers
12 views

$d \log f'$ is a compactly supported radon measure for $f\in C_c^{1}$ with bounded variation

In a note of John N. Mather, "Commutators of Diffeomorphisms, III: a group which is not perfect," he says that it is well known that if $f$ is a compactly supported $C^1$ diffeomorphism on ...
3
votes
1answer
73 views

Absolute continuity and sample paths of Brownian motion

An offhand remark in Morters and Peres' book on Brownian motion says that Brownian motion is a.s. absolutely continuous on compact intervals (see page 147, immediately preceding the statement of ...
1
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1answer
128 views

Function f is bounded on $[0, \infty)$

Let $f : [0, \infty) \to R$ be continuous such that $lim_{x \to +\infty} f(x) = 0$. How can I Prove that f is bounded on $[0, \infty)$. I know that condition for a function to be bounded is - ...
1
vote
1answer
46 views

Compute total variation when discontinuities are given bounds

Say you have a function such as $f(x)=1+\sin(x)$ that is defined from $-\frac{\pi}{4}$ to $\frac{\pi}{4}$. Everywhere else, the function takes on the value $-\frac{1}{2}$. How do you compute the ...
0
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1answer
50 views

Calculate total variation of g on a given interval.

I am dealing with the following function: $$g(x) = \left\{ \begin{array}{lr} 1+\sin(x) & -\frac{\pi}{4} < x < \frac{\pi}{4} \\ -\frac{1}{2} & otherwise ...
2
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1answer
73 views

a question about proving a normed space is complete

Let [a,b] be an interval in R,and denote by E the Vector space of functions f:[a,b]->R such that f is of bounded variation over [a,b] and f(a)=0.Prove that by setting $||f||=Var|_{a}^{b}(f)$ for each ...
0
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1answer
86 views

Understanding bounded variation

In my analysis course we are covering the topic of bounded variation fuctions and I am really having a very hard time trying to get the concept. My main problem is that I don't get how can a function ...
1
vote
2answers
32 views

Establishing a bound in the complex plane

so my function is $$f(z)= \frac{e^{iz}}{z^2+a^2} $$ What is getting to me and probably I should've been comfortable with this fact is how they establish this upper bound: $$\bigg|\int_{0}^{\pi} ...
4
votes
1answer
444 views

Monotone increasing continuous function with $\int_a^b f' = f(b) - f(a)$ which is not absolutely continuous

If $f:[a, b] \to \mathbb{R}$ is continuous and real-valued, f' integrable on [a, b], and $\int_a^b f' = f(b) - f(a)$, must f be absolutely continuous? What if f is monotone increasing? For the ...
1
vote
0answers
119 views

Absolutely continuous iff continuous of bounded variation

I have the following problem (taken from q1 p341 of Kolmogorov and Fomin's Introductory Real Analysis), which I am struggling to prove completely. I think I know how to show the only if part, but not ...
0
votes
1answer
83 views

Showing that functions of bounded variation are not closed under composition

Find functions $g: [a, b] \to [c, d]$ and $f: [c, d] \to \mathbb{R}$ both of bounded variation, with f continuous, so that $f \circ g$ is not of bounded variation. This occurs as a request for a ...
2
votes
1answer
34 views

$\int_0^\pi g(x)\sin(nx) dx$ coverges to $0$ for any function $g$ is bounded variation

I have a homework as follows : Prove that $\int_0^\pi g(x)\sin(nx) dx$ coverges to $0$ for any function $g$ is bounded variation on $[0,1]$. my attempt: for any bounded variation function $g$, ...