Questions tagged [bounded-variation]

For questions about functions $f$ defined on an interval $[a,b]$ such that there exists a constant $M>0$, such that if $a=x_0<x_1<\ldots<x_n=b$, $n\in\mathbb N^*$, then we have $\sum_{k=1}^n|f(x_k)-f(x_{k-1})|\leq M$. This concept can be generalized to infinite intervals, requiring that the constant is uniform.

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Estimating integral expression by using total variation of an integrand term

Consider the following integral expression: $$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/3} f(x) f(y) \frac{(g(x)-g(y))(x-y)}{|x-y|^{3}} d x d y $$ for $\epsilon>0$, $f \in L^\infty(\mathbb R)$,...
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Are the right and left hand limits of an increasing function locally uniformly convergent?

Let $f : \mathbb{R} \to \mathbb{R}$ be increasing. Hence we know $f$ is differentiable almost everywhere, and that the right and left hand limits, respectively $f(x+)$ and $f(x-)$, exist at every $x \...
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Power rule for functions of bounded variation

Let $f : \mathbb{R} \to \mathbb{C}$ be a function of normalized bounded variation (NBV), meaning that $f$ is of bounded variation, $f$ is right continuous, and $f(x) \to 0$ and $x \to -\infty$. As ...
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A continuous function with bounded variation which isn't monotonic.

so I got a homework assignment to prove a function is continuous with bounded variation but isn't monotonic on any subinterval. The function is defined as, let: $$g\left(x\right)=\left|x\right|\ x\in\...
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Convergence of Riemann-Stieltjes integral of square function.

I got this question which I think I know the solution to but I am not certain I am doing it right, will gladly use some help about it. So the question is like that, assume we have function $\alpha$ ...
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  • 33
2 votes
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Continuity + finite number of local maxima and minima implies absolute continuity

I'm going through a set of older exams of a real analysis course to practise for the exam and I cannot prove the following: Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function with a finite ...
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3 votes
2 answers
195 views

If $f\in C([a,b])$, $V(f^2;[a,b])<\infty$, then $V_2(f;[a,b]) \leq C(V(f^2;[a,b])+\|f\|_\infty^2)$

Assume that $f\in C([a,b])$ is a complex-valued function with $V(f^2;[a,b])<\infty$. Prove that $$V_2(f;[a,b])\leq C(V(f^2;[a,b])+\|f\|_\infty^2),$$ where $C>0$ is independent of $a,b$ and $f$. ...
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Jordan decomposition of $F \in BV$

Suppose $F \in BV$, then its Jordan decomposition is given by $F = \frac{1}{2}(T_F + F) - (T_F - F)$ where $T_F$ is the total variation of $F$. In Folland's text he says since $x^+ = \max(x, 0) = \...
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  • 1,366
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Complex valued Stieltjes integrals : If $f\in\mathcal{R}(\alpha)$, do we have $\mathfrak{Re}(f), \mathfrak{Im}(f)\in\mathcal{R}(\alpha)$?

Given a bounded complex function $\alpha:[a,b]\to\mathbb{C}$, we can define the Riemann-Stieltjes integral of $f:[a,b]\to\mathbb{C}$ (also bounded) in a way that is very much analogous to the usual ...
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Total variation of $F:[a,b]\to\mathbb{R}^{N}$ where each component of $F$ is given by a definite integral

Let $\alpha:[a,b]\to\mathbb{R}$ be monotone increasing and let $\mathcal{R}(\alpha)$ be the set of all functions $f:[a,b]\to\mathbb{R}$ that are Riemann-Stieltjes integrable with respect to $\alpha$. ...
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Does pointwise convergence imply bounded variation?

Problem: Let $\{f_k\} \in BV([a,b])$ and $f_k \rightarrow f$ pointwise in $[a,b]$. Suppose $V(f_k;a,b) \le M$ for all $k \in \mathbb{N}$ for some $M \in (0,\infty)$. Is it enough for a pointwise ...
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If $f(x)=-e^{-x}$ on $(0,\infty)$, Prove that $\inf_{x>0}$ $f(x)=-1$.

If $f(x)=-e^{-x}$ on $(0,\infty)$, Prove that $inf_{x>0}$ $f(x)=-1$. For all $x>0$, $f(x)>-1$ then $-1$ is a lower bound of $f$ on $(0,\infty)$. I did it up to here. And I want to find a any $...
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Prove that $f(x)=\frac{1}{x}$, $g(x)=x$ are unbounded on $(0,\infty)$.

Prove that $f(x)=\frac{1}{x}$, $g(x)=x$ are unbounded on $(0,\infty)$. We show that $f$ is unbounded by assuming that there is a bound $M>0$ and then arriving at a contradiction. From the graph of $...
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1 answer
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One question about f is a continuous function of bounded variation

$f$ is a is continuous and has bounded variations, then for every sequence $Π^n$ of subdivisions of $[0,T]$ such that $|Π^{n}|:=sup |t_i - t_{i-1}| → 0$ then how to prove that $$\lim \sup_n \sum^n_{k=...
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1 vote
0 answers
27 views

Sum of a locally BV right-continuous function and a locally BV left-continuous function

Let $f:\mathbb{R}\to\mathbb{R}$ be a right-continuous function that is locally of bounded variation (i.e., of bounded variation on every $[a,b]\subset\mathbb{R}$; henceforth, "LBV function"),...
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If $x_1, x_2, x_3, x_4 > 0$ and $x_1+x_2+x_3+x_4 =2$ , then $P = (x_1 + x_2) (x_3 + x_4)$ is bounded between?

If $x_1, x_2, x_3, x_4 > 0$ and $x_1+x_2+x_3+x_4 =2$, then $P = (x_1 + x_2) (x_3 + x_4)$ is bounded between, A. 0 and 1 B. 1 and 2 C. 2 and 3 D. 3 and 4 How do you solve these kinds of questions? I ...
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6 votes
1 answer
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monotone functions agreeing with Holder functions on a large set

Let $\alpha \in (0,1)$, $f:[0,1]\rightarrow \mathbb{R}$ be a continuous monotone function and $\varepsilon>0$. Does there exist a function $\phi_{\varepsilon} \in \mathcal{C}^{\alpha}$ such that $\...
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Why can we assume WLOG $\alpha$ is increasing?

I have a question regarding the proof of Theorem 9.8 from Mathematical Analysis by Tom Apostol below: Theorem 9.8: Let $\alpha$ be of a bounded variation on $[a,b]$. Assume that each term of the ...
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2 votes
3 answers
72 views

Does the derivative of a vector-valued BV function $f(x)$ equal to the norm of $f'(x)$?

Let $f(x)$ be a real-valued function on $[a,b]$ of bounded variation. It is standard that $f(x)$ is almost everywhere differentiable, and that $\dfrac{{\rm d}}{{\rm d}x} V^x_a f = |f'(x)|$ for a.e. $x\...
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Truncation argument to show that $\mathcal{H}^{N-1}(\partial \Omega) < \infty$ implies that $\Omega$ has finite perimeter

I am currently reading "Functions of Bounded Variation and Free Discontinuity Problems" by Luigi Ambrosio, Nicola Fusco, Diego Pallara. I am stuck with Proposition 3.62: Any open set $\...
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1 answer
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Measurability of level set mapping

I want to show that the mapping $\mathbb{R}\ni t\mapsto \mathcal{H}^{n-1}(\Omega \cap \{x:u(x)>t\}) = \sup\left\{\displaystyle\int_{\{x:u(x)>t\}}div(v)\,dx, v=(v_1,...,v_n)\in C^{\infty}_0(\...
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1 vote
1 answer
89 views

Total variation on $\mathbb{R}$

The total variation of differentiable function $f$ on the closed interval $[a,b]\subset\mathbb{R}$ is given by $$V_a^b(f)=\int_a^b|f'(x)|dx.$$ Does the same formula hold for the total variation of ...
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Is there a function on $[a,b]$ that has a bounded derivative on $(a,b)$, is NOT continuous at $a$ and $b$, and is NOT of bounded variation on $[a,b]$?

It is well known that a continuous function on a compact interval $[a,b]$ that has a bounded derivative on $(a,b)$ is of bounded variation on $[a,b]$. I am curious that whether the continuity at the ...
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Poincare type inequality for BV functions

I want to show $\|u-\overline{u}\|_{\frac{n}{n-1},\,B(r)} \leq C(n) \| Du\|(B(r))$ with $\overline{u}$ the mean value on $B(r)$ for $u\in BV(\mathbb{R}^n)$. I already have shown that $\|u\|_{\frac{n}{...
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1 vote
0 answers
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Outward pointing vector on Lipschitz boundary

I have some questions on Lipschitz domains and their unit outward pointing vectors. My questions are listed below, I would appreciate direct answers and/or references on the subject. What is the good ...
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0 votes
1 answer
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A continuous bounded variation function in $[0,1]$ that is absolutely continuous in $(a,1]$, but is not in $[0,1]$

I am seeking for a continuous of bounded variation function in $[0,1]$ that is absolutely continuous in $(a,1]$ for all $a\in(0,1)$, but is not in $[0,1]$. The function $x\sin\left(\frac{1}{x}\right)$,...
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1 vote
1 answer
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Is $f(x)=x\sin(1/x)$ with $f(0)=0$ of bounded variation on $[0,1]$? - Problem with abs. continuous

I am having the following trouble: From Is $f(x)=x\sin(\frac{1}{x})$ with $f(0)=0$ of bounded variation on $[0,1]$?, $x\sin(1/x)$ has not bounded variation in $[0,1]$. $x\sin(1/x)$ has derivative $-\...
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0 votes
1 answer
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A function of bounded variation in a regular set has bounded variation in $\mathbb R^N$ and a formula for its variation

Let $\Omega \subset \mathbb R^N$ be an open set. The total variation of a function $u \in L^1(\Omega)$ in $\Omega$ is given by $$ |Du|(\Omega) := \sup\left\{\int_\Omega u \ \text{div}\varphi \ dx \ : \...
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bounding $p$ variation with dyadic intervals

Let $T>0$ be finite and $x \colon [0,T] \to \mathbb{R}^d$ be a continuous path. Let $x_{s,t}:= |x(t)-x(s)|$. For each $m= 1,2,\dots$, let $t^m_j:= \frac{jT}{2^m}$. These points are called dyadic ...
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Bounded bivariation & bounded from zero doesn't imply that reciprocal is of bounded bivariation

Let $I, J \subset \mathbb{R}$ be compact intervals. We call $f\colon I \times J \to \mathbb{R}$ is of bounded bivariation if for every partition $\{x_0, \ldots, x_m\}$ of $I$ and $\{y_0, \ldots, y_n\}$...
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0 votes
1 answer
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Showing Riemann-Stieltjes Integrability

I am doing an independent study in Measure Theory, and I am using Measure and Integral by Wheeden and Zygmund. In chapter 2, "Functions of Bounded Variation and the Riemann-Stieltjes Integral&...
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1 answer
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a product of functions of bounded variation is a function of bounded variation using Jordan's theorem

We proved in class today that if $f ,g : [a,b] \rightarrow \mathbb{R}$ are functions of bounded variation then so if $fg$. It was done directly using definitions and it was rather lengthy in my ...
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2 votes
1 answer
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Trying to imagine how functions of bounded variation look like

I am trying to wrap my head around what functions of bounded variation mean or how they exactly look like. On Wikipedia I read that For a continuous function of a single variable, being of bounded ...
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1 vote
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Q : Let $f : \mathbb Z\to \mathbb Z$ such that, for all $x,y\in\mathbb Z:$ $f(f(x) − y) = f(y) − f(f(x)).$ Show $f$ is bounded.

I came to the conclusion that f is periodic with period |f(x0)| for x0 non-zero. But I don't see how the periodicity in domain translates to a bound in the codomain. It probably has something to do ...
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3 votes
2 answers
105 views

An application of Baire Category Theorem

I am trying to prove a proposition that $BV[a.b]\cap C[a.b]$ equipped with the $||\cdot||_\infty$ is Baire 1 category set, which will tell us that $E=\{f:V(f)=\infty, f\in C[a,b]\}$ is a dense Baire 2 ...
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1 vote
1 answer
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Variation of a function on a countable subset

Let $E$ be a $\mathbb R$-Banach space, $g:[0,\infty)\to E$, $$\operatorname{Var}_\varsigma g:=\sum_{i=1}^k\left\|g(t_i)-g(t_{i-1})\right\|_E$$ for $\varsigma=(t_0,\ldots,t_k)\in\mathcal S_I$, where $$\...
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2 votes
1 answer
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Clarification on bounded variation of the terms in the Ito's formula as per Ikeda and Watanabe's book

Let $X(t) = X(0) + M(t) + A(t)$ be a continuous semi-martingale where $M \in \mathscr {M}$ and $A \in \mathscr A.$ Let $F: \mathbb R \to \mathbb R$ be $C^2$ be a function of class $C^2$. Then $$F(X(t)...
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3 votes
1 answer
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bounded variation functions on $[0,1]$ are always $L_2[0,1]$?

My question is $BV[0,1] \subset L_2[0,1]$ or not. My own answer (not sure correct): $f \in BV[0,1]$ implies that removing discontinuity, we have continuous function $\tilde{f}$ such that \begin{align*}...
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9 votes
0 answers
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Can I split $E$ in equal volume parts?

Problem: Let $E \subset \mathbb{R}^N$ be a connected, bounded, open and smooth (or just $N$-measurable) set and denote with $\mathcal{L}^N$ the Lebesgue measure. Define $\Omega_i = \{ x \in \mathbb{R}^...
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3 votes
1 answer
86 views

Is $BV$ the dual of a separable Banach space?

In literature I am reading right now, it says $BV(\Omega)$, where $\Omega \subset \mathbb{R}^n$ an open bounded set, is the dual of a separable space. Is it a dual separable Banach space, or not a ...
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1 vote
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Variation of g, $\Vert f \Vert$, being less than variation of $\hat g$ on page 49 of Schechter's Principle of Functional Analysis

I am confused by how $V(g) = \Vert f \Vert ≤ V(\hat g),$ with $f$ being a bounded linear functional, come about via (2.48) and (2.49) on page 49 of Schechter's Principle of Functional Analysis. First ...
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Does boundedness in $L^\infty ( 0,T; L^1 (\Omega))$ imply boundedness in $L^\infty (0,T;BV(\Omega))$?

Let $\Omega$ be an open bounded domain in $\mathbb{R}^n$ and let $T>0$. Assume $\lVert \nabla u \rVert_{L^\infty ( 0,T; L^1 (\Omega))} \leq C_1$. Does this imply $\lVert u \rVert_{L^\infty (0,T;BV(\...
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  • 73
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Solution of PDE (Existence)

Recently I came acroos the following PDE: $$ \partial_t u+\nabla \cdot f(u) = 0, ~ u(0, x) = u_0(x) ~~~(\text{on }(0, T) \times \mathbb{R}^d) $$ We assume $f: \mathbb{R} \rightarrow \mathbb{R}^d$ to ...
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Proof that Functions of Bounded Variation are Differentiable Almost Everywhere

I'm working through a proof in my measure theory class that a function $F : [a,b] \to \mathbb{R}$ of bounded variation is differentiable almost everywhere. We first prove that $F$ can be written as $G ...
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  • 1,857
2 votes
3 answers
109 views

Prove that there exist $M$ such that when $|h|$ is sufficiently small, $ \frac{1}{h}\int_a^b|f(x+h)-f(x)|\mathrm{d}x\leq M. $

Suppose $f(x)\in$ BV$([a,b])$(bounded variation). Prove that there exist $M$ such that when $|h|$ is sufficiently small, $$ \dfrac{1}{h}\int_a^b|f(x+h)-f(x)|\mathrm{d}x\leq M. $$ By Jordan ...
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  • 1,115
1 vote
0 answers
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Compact immersions of the bounded variation space $BV(\Omega)$

I have missed a detail attending a lesson, I would like at least some references to fill my gap. We were proving the compact immersion of $BV(\Omega)$ in $L^p(\Omega)$ with $1\leq p<1^{*}$ and $\...
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2 votes
0 answers
34 views

Construct counterexamples regarding bounded variation function

Construct a function $f(x)\in C(0,1)$ such that $f'(x)$ exists almost everywhere and $f'(x)$ is Lebesgue integrable while $f\notin $ BV([$0,1$]). Construct a function $f(x)\in C(0,1)$ such that $f'(x)=...
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  • 1,115
2 votes
0 answers
48 views

Total Variation of a Borel Measure vs Total Variation of a Function

Let $\nu$ be a finite signed Borel measure on the interval $[a,b]$, and let $F_\nu$ be the function given by $F_\nu(x) = \nu([a,x])$. I'd like to show that the total variation $V(F_\nu, [a,b])$ is the ...
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  • 1,857
2 votes
0 answers
48 views

Jordan decomposition of differentiable functions

Let $f:[a,b]\to \mathbb{R}$ be a function of bounded variation. A well known result, known as Jordan's decomposition, states that we can write $f=W_1-W_2$ where $W_1,W_2$ are increasing and minimal in ...
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0 votes
2 answers
64 views

Lipschitz does not bound increments by 1-variation

Can you find a counterexample to the below claim? Claim: Let $f:\mathbb{R}\to \mathbb{R}$ be Lipschitz, i.e. $$|f(x)-f(y)| \leq K_1 | x - y |$$ for all $x,y$ . Let $x, y :[0,T]\to \mathbb{R}$ be ...
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