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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2answers
50 views

The space $BPV(0,1)$ is a separable metric space under certain metric.

This is exercise 2.42 from Leoni's book A First Course in Sobolev Spaces. The BPV is defined as the space contain the function $u$ such that $$ \text{Var}[u]:=\sup\left\{ \sum_{i=1}^n|u(x_i)-u(x_{i-...
2
votes
2answers
34 views

$W^{1,1}\subseteq AC$ and a certain property implies BV: why?

Brézis states that the functions in $W^{1,1}(I)$, with $I$ a bounded interval, are absolutely continuous, and that, for $u\in L^1(I)$, if the following holds for some constant $C$: $$\left|\int_Iu\...
0
votes
0answers
21 views

Reconstruction of a function of bounded variation

The variation of a function $f:[a,b]\to\mathbb R$ is defined by \begin{align*} \text{Var}(f,[a,b]):=\sup_P\sum_{j=1}^n|f(t_{j-1})-f(t_j)|, \end{align*} where $P$ runs through all partitions $P=(a=t_0&...
0
votes
0answers
28 views

The $L^2$ convergence of semi-$p$-lapace equation

This question is similar to the one I post early here. But this one might be more reasonable I think... Let $g\in L^\infty(\Omega)$ be given, where $\Omega\subset \mathbb R^2$ is open bounded with ...
1
vote
0answers
28 views

The convergence of $p$-laplace equation

Let $g\in L^\infty(\Omega)$ be given, where $\Omega\subset \mathbb R^2$ is open bounded with smooth boundary. Define, for $1<p\leq 2$, $$ u_p:=\operatorname{argmin}\left\{\int_\Omega|u-g|^pdx+\...
2
votes
3answers
2k 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 ...
1
vote
0answers
16 views

Bounded piecewise monotone function is of bounded variation?

Suppose that $f:[a,b]\rightarrow\mathbb{R}$ is bounded by an $M>0$ and there exist points $a=x_0<x_1<\ldots<x_N=b$ such that $f|_{[x_{j-1},x_j)}$ is monotone for $j=1,\ldots,N$. I want to ...
3
votes
1answer
41 views

Is $AC[a,b]$ closed in $(BV[a,b],TV)$?

Consider $BV[a,b]$ the space of all bounded variation functions on a real interval $[a,b]$, endowed with the total variation norm $TV$. $AC[a,b]$, the space of absolutely continuous functions, is a ...
0
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0answers
47 views

Total variation on BV functions: “Banach seminorm”?

Suppose I consider the space $BV[a,b]$ of all bounded variation functions on $[a,b]$ a real interval. I endow it with $\|f\|=TV(f)$ the total variation norm. Do I get a Banach space? How can I prove ...
0
votes
1answer
36 views

Limit of the integral of $f(\epsilon t) g(t)$ as $\epsilon\to 0$, when $f$ has bounded variation

I am needing such a "result", which I do not know if it is true, for another question, Convolution with Gaussian question. Let $f\in L^\infty(\mathbb R)$ be of bounded variation on any interval $[a,b]...
0
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1answer
24 views

a problem concerning continuous functions of bounded variation [duplicate]

Here is a problem: Suppose $f,g: [a,b]\rightarrow \mathbb{R}$ are both continuous and of bounded variation. Show that the set $\{(f(t),g(t))\in\mathbb{R}^2: t\in [a,b]\}$ CANNOT cover the entire unit ...
0
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0answers
7 views

How to indicate which is the best bound for accuracy either the max or min bounds?

A Few lists of precipitation data (P-data at different stations) in descending order is used to estimate streamflow at its corresponding stations. For every P data, 2 streamflow values are ...
0
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0answers
13 views

Proof Function is Bounded/Unbounded

How can I prove that the function $$\sigma_i\left(t\right) = k_i\left[\left(a+b\left(T_i-t\right)\right)e^{-c\left(T_i-t\right)}+d\right]$$ is bounded/unbounded? Note: $\sigma_i\left(t\right)$ is ...
1
vote
1answer
24 views

Prove absolute continuity without Banach-Zarecki

Let $f$ be a real-valued continuous function of bounded variation on $[a,b]$. Suppose $f$ is absolutely continuous on $[a+\eta,b]$ for every $\eta\in(0,b-a)$. Show that $f$ is absolutely continuous on ...
0
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0answers
26 views

Minimum surface area soap film variational principles

An axisymmetric soap film $y(x)$ is formed between two circular wires at $x = ±l$. The wires both have radius $r$. Show that the shape that minimises the surface area takes the form $$y(x) = ...
3
votes
0answers
228 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. ...
0
votes
1answer
123 views

how can you “fix” one of the definitions of a BV function of one variable?

Math people: My question is similar to that in Two definitions of "Bounded Variation Function" . If you look at that question, you will notice that people treat definitions (1) and (2) ...
-3
votes
1answer
61 views

$\sin(1/x)$ is not BV. [closed]

I have to prove that $$ \begin{cases} \begin{array}{cc} \sin(1/x) & x\in \Big(0,\frac{2}{\pi}\Big] \\ 0 & x=0 \end{array} \end{cases} $$ is not of bounded variation. I ...
0
votes
1answer
40 views

Find the 'rough' error bound to the composite simpson rule

Provide a rough error bound for the following composite simpsons rule. I am aware that the upper bound is $f$ to the forth derivative evaluated at some $t$ in the open interval $(a,b)\frac{h^4(b-a)}{...
0
votes
0answers
19 views

Characterization of the Jordan decomposition of a real-valued function of bounded variation from Folland.

This is a characterization of the Jordan decomposition of $F$ from Folland's Real Analysis. However, I can't see how the characterization makes sense. Let $F\in BV$ be a real valued function and $T_F$...
1
vote
0answers
40 views

Functions of bounded variations and Riemann-Stieltjes integral

In my work, I used recently the classical Riesz theorem. It has lead me to study functions of bounded variations and Riemann-Stieltjes integrals. Unfortunately, even if there exist a lot of books and ...
4
votes
0answers
66 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 for this problem: Show that $\| f \|_{BV} = | f(a) | + V_{a}^{b} f$ defines a norm in the space $BV[a,b]$. My ...
4
votes
4answers
57 views

Function is bounded above [duplicate]

Is there a good way to show that $\frac{\sin(x)}{x}$ is bounded above by $1$? We can see visually that $\frac{\sin(x)}{x}$ is bounded above by $1$ because the tallest hump is at the origin and $\lim_{...
0
votes
0answers
34 views

Prove that if $f$ is of bounded variation on $[a, b]$, then $f$ is bounded on $[a, b]$.

Prove that if $f$ is of bounded variation on $[a, b]$, then $f$ is bounded on $[a, b]$. My professor proved the proposition like the following processes: Choose $x$ between $a$ and $b$, that is, $...
3
votes
0answers
23 views

Is it okay if I considered $f$ as uniformly continuous and bounded on specific interval if $f$ is continuous?

Problem Suppose $f$ is continuous and $\phi$ is of bounded variation on $[a, b]$. Show that the function $\psi(x)=\int_a^xfd\phi$ is of bounded variation on $[a, b]$. If $g$ is continous on $[a, b]$...
0
votes
1answer
18 views

Please explain to me the following.

1)How does the graph of a funtion of bounded variation behave. 2)Why a bounded function is not always a function of bounded variation.Please explain graphically. 3)What purpose does bounded variation ...
1
vote
0answers
21 views

Can someone explain the detail of the proof (rectifiable curve)

Curve $C$ in the plain, \begin{align} C:= \begin{cases} x=\phi(t) \\ y=\psi(t) \end{cases} \quad, \quad a \le t \le b \end{align} The graph of C is $\{(x, y): x=\phi(t), y=\psi(t), a\le t\le b\}$ $\...
1
vote
1answer
65 views

What is the name of this theorem? I do not know how to type some symbols in google.

In my textbook, the following theorem is proved. But I do not understand it. So I am finding some other documents make me understand the theorem. Could someone let me know this theorem's name? Since I ...
2
votes
0answers
67 views

Pointwise convergence of a sequence of approximate limits of BV functions.

So, let $\Omega\subset \mathbb R^2$ bounded and consider a sequence of functions $\{u_k\}_{k\in\mathbb N}\subset BV(\Omega)$ and $u\in BV(\Omega)$ such that $u_k\rightarrow u$ weakly* in $BV(\Omega)$. ...
0
votes
0answers
47 views

Can someone prove this bounded variation problem?

I am doing my homework, which is solving all even number problems in textbook. Finally, I have almost finished.... but I failed to solve one problem. I am not sure that it is okay if I post the whole ...
1
vote
1answer
58 views

How to prove that f/g is bounded variation?

I am trying to prove that when $f$ and $g$ are of bounded variation on $[a, b]$, $f/g$ is of bounded variation on [a, b] if there exists an $\varepsilon\gt 0$ such that $|g(x)|\ge \varepsilon$ for $x\...
2
votes
1answer
49 views

$f'$ exists for a function of bounded variation

If $f \in BV[a, b]$, show that $f'$ exists and is integrable. My Attempt : I know that for any $f \in BV[a, b]$, we can write it as difference of two monotonic increasing functions and monotonic ...
6
votes
3answers
106 views

Necessary and sufficient condition for a curve to have infinite length

What is the necessary and sufficient condition for a curve to have infinite length in a compact interval? Say the curve is restricted to $[0, 1]$. I vaguely remember that it is related to the ...
1
vote
1answer
17 views

Is $H^1$ subspace of Sobolev space $W^{1,1}$?

Let $I \in \mathbb R^d$. A result I need, states that a certain property holds weakly in $BV(I)$, and holds strictly in $W^{1,1}(I)$ (which is a subspace of $BV(I)$). I would actually need this ...
1
vote
2answers
72 views

A function with rectifiable graph satisfies Lipschitz condition on a large set

Let $f$ be real-valued on $[0,1]$. Let $G$ be the graph of $f$, and suppose it is rectifiable. Let the length of $G$ be $L$. Put $\epsilon > 0$. Then there exists a positive constant $c$ and a ...
0
votes
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 ...
4
votes
1answer
50 views

How to show that $BV[0,1]$ ( the space of all functions on $[0,1]$ of bounded variation ) is not complete under supremum norm?

How to show that $BV[0,1]$ ( the set of all functions of bounded variation ) is not complete under supremum norm , by explicitly constructing a Cauchy sequence which does not converge or by ...
0
votes
2answers
77 views

Real Analysis, Folland problem 3.5.27 Functions of Bounded Variation

Background Information: If $F:\mathbb{R}\rightarrow \mathbb{C}$ and $x\in\mathbb{R}$, we define $$T_{F}(x) = \sup\{\sum_{1}^{n}|F(x_j) - F(x_{j-1})|:n\in\mathbb{N},-\infty<x_0<\ldots<x_n = x\...
0
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0answers
29 views

Construct a continuous function $f$ with $0\leq f\leq 1$ such that $\int_a^bf d\alpha\geq\alpha(d)-\alpha(c)-\epsilon$

Suppose that $\alpha$ is right continuous and increasing. Given $\epsilon > 0$ and $[c,d]\subset[a,b]$, construct a continuous function $f$ with $0\leq f\leq 1$ such that $\int_a^bf d\alpha\geq\...
1
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0answers
39 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 ...
1
vote
1answer
80 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. ...
0
votes
1answer
36 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 $|f^\...
0
votes
1answer
73 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 ...
3
votes
1answer
77 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 ...
1
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0answers
46 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, \...
2
votes
1answer
71 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, ...
3
votes
2answers
182 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 $V^...
0
votes
0answers
38 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 ...
0
votes
0answers
26 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 $V_{a}^{b}(X_\mathbb{Q}) = +\infty$ ...
1
vote
0answers
46 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 $[a,b]$. $f_1=\frac{1}{2}(v(x)...