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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1answer
22 views

When is the Stieltjes integral of bounded variations?

I was trying to figure out when a Riemann or Lebsgue Stieltjes integral is of bounded variation. For simplicity let $f$ be a increasing RCLL function; when is that $$\int_0^t g(x) df(x)$$ is of ...
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1answer
41 views

Decomposition of function of bounded variation

Suppose we have $f:\mathbb{R} \rightarrow \mathbb{R}$ which is of bounded variation. I would like to show that it can be presented as a sum of left and right continuous functions of bounded variation. ...
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1answer
26 views

Bound the variation of this decreasing function

Let $f(x)$ be a decreasing function defined over the interval $[0,a]$, with $f(0)=b$. The first derivative of $f(x)$, which is negative, is such that $f^\prime(x) > g(x)$, or equivalently ...
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1answer
51 views

Construct a non-monotone continuous function of bounded variation

Construct a continuous function of bounded variation on $[0,1]$ which is not monotone in any subinterval. We can follow the pattern of the Cantor-Lebesgue function (somewhat). For example, at the ...
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0answers
44 views

Is it possible to upper bound conditional expectation expression: $\mathbb E[X | X > c] - c$?

As the titles suggests, I am trying to see if we can upper bound $\mathbb{E}[X \text{ }| \text{ }X > c] - c$ For now, I am assuming bounded mean on both sides: $0 < m \leq \mathbb{E}[X] \leq ...
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1answer
33 views

Does differentiability imply having bounded variation on some subinterval?

Suppose that $f:(a,b)\to\mathbb{R}$ is a differentiable function. Does it follow that $f$ has bounded variation on some subinterval $[c,d]\subset (a,b)$? Details and ideas Being differentiable ...
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0answers
37 views

Wrong result: a continuous function has zero $p$-variation, for every $p$. Where's the error?

Let $\Pi_n$ be a sequence of partitions with $|\Pi_n| \to 0$. Then the $p$-variation of a continuous function $g$ along the partitions $\Pi_n$ is defined as $$V_T^p(g) = \lim_{n \to \infty} V_T^p(g, ...
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0answers
30 views

Given $f(x) = x^{1/3}$, show that $f \in BV[0, 1]$

I'm learning about functions of bounded variations and need some help with this problem: Given $f(x) = x^{1/3}, x \in [0, 1]$ show that $f \in BV[0, 1]$. My work and thoughts: We know that ...
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0answers
24 views

Show that $V_{a}^{b}(X_\mathbb{Q}) = +\infty$ on any interval [a, b]

I'm self-studying the theory of functions of bounded variations from the book of Neal Carother (Real Analysis) and need some help with this problem: Show that $V_{a}^{b}(X_\mathbb{Q}) = +\infty$ ...
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2answers
156 views

If $f$ is of bounded variation on $[a+\epsilon, b]$, does it imply $f$ is of bounded variation on $[a,b]$?

The problem goes like: Suppose that $f\in B[a,b]$. If $V^b_{a+\epsilon}f\leq M$ for all $\epsilon >0$, does it follow that f is of bounded variation on $[a,b]$? I think the answer is yes. Since ...
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0answers
70 views

Upper bound on successive difference inequalities

I would like to know tight upper bounds of the following equations perhaps some might have telescopic sum which can result in very tight bound.In the following equations assume $B\geq a_i \geq ...
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0answers
35 views

Relationship between functions of bounded variation and signed measures

Let $f$ be a left continuous function of bounded variation on $[a,b]$ s.t. $f(a)=0$. $f$ can be written as the difference $f=f_1-f_2$ of two non-decreasing functions on $[a,b]$. ...
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1answer
37 views

If $\| f_n - f \|_{BV} \rightarrow 0$, then $f_n$ converges uniformly to $f$ on $[a, b]$

I'm learning about functions of bounded variation and need help with this theoretical problem: Let $f_n : [a, b] \rightarrow \mathbb{R}$ a sequence of functions in $BV[a, b]$. Show that if $\| f_n ...
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0answers
39 views

$\| f \|_{BV} = | f(a) | + V_{a}^{b} f$ defines a norm in the space $BV[a,b]$

I'm learning about functions of bounded variations and need to verify my work this problem: Show that $\| f \|_{BV} = | f(a) | + V_{a}^{b} f$ defines a norm in the space $BV[a,b]$. My attempt ...
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1answer
36 views

Exploring the total variation of a $C^1$ function

We define the Banach space of functions of bounded variation on $\Omega\subseteq\mathbb{R}^n$ (assume as smooth a domain as we need) as all $u\in L^1(\Omega)$ for which ...
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1answer
122 views

Understanding Lemma: $\left\lVert f \right\rVert_{\infty} \leq \left|f(a) \right| + V_{a}^{b} f$

I'm learning about functions of bounded variations and need help to understand the proof of this lemma: Lemma. If $f : [a,b] \rightarrow \mathbb{R}$ is of bounded variation, then f is also ...
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1answer
32 views

Are there functions that are Holder continuous but whose variation is unbounded?

I have recently been introduced to the concept of Holder condition and I was told that there are functions that are Holder continuous but whose variation in unbounded. Can anyone present an example, ...
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1answer
41 views

Let $f$ be of bounded variation on $[a,b]$, and define $v(x) = f_{[a,x]}$. show $\int_a^b |f'|\leq TV(f).$

Let $f$ be of bounded variation on $[a,b]$, and define $v(x) = TV(f_{[a,x]})$ for all $x \in [a,b]$. show that $|f'| \leq v'$ a.e. on $[a,b]$, and infer from this that $$\int_a^b |f'|\leq TV(f).$$ ...
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0answers
38 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 $Re(f(z))<0$ ...
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1answer
32 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 ...
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1answer
18 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 ...
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1answer
18 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
22 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 ...
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27 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 $f$ ...
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1answer
22 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: ...
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0answers
10 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 ...
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2answers
33 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
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3answers
98 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 ...
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1answer
38 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 ...
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0answers
22 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 if ...
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0answers
18 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
35 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 ...
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2answers
58 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
23 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
67 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 ...
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1answer
20 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 ...
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0answers
14 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 ...
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1answer
19 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 $$ \|u_\epsilon-u_0\|_{L^2}^2+\epsilon |u_\epsilon|_{TV(\Omega)}\to 0 $$ ...
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0answers
43 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
30 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
49 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
70 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
154 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
42 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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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 = ...
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1answer
28 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 ...
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0answers
22 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
74 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
10 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_{ ...