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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3
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0answers
199 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
122 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
42 views

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

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 ...
-2
votes
0answers
23 views

Proving the total variation of $f$ is $\int_a^x|f'(t)|\;dt$ [duplicate]

Let $f:[a,b]\to\mathbb{R}$ be differentiable in $(a,b)$ with continuous derivative $f'$ on $[a,b]$. Prove that $f$ is of bounded variation in $[a,b]$ and $$V(f,[a,x]) = \int_a^x|f'(t)|\;dt, $$ where ...
0
votes
1answer
35 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 ...
0
votes
0answers
14 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 ...
1
vote
0answers
25 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
60 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
55 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 ...
0
votes
0answers
27 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
20 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, ...
1
vote
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
19 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
63 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
44 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
49 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 ...
2
votes
1answer
38 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
98 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
16 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
59 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
27 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
37 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
65 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 = ...
0
votes
0answers
28 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 ...
1
vote
1answer
37 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
58 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
33 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 ...
0
votes
1answer
66 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 ...
4
votes
1answer
51 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
vote
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
49 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
173 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 ...
0
votes
0answers
35 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
25 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
42 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 $[a,b]$. ...
0
votes
0answers
74 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 ...
-1
votes
3answers
3k views

Upper Bound of Logarithm

For $1\leq x < \infty$, we know $\ln x$ can be bounded as following: $\ln x \leq \frac{x-1}{\sqrt{x}}$. Then what is the upper bound of $\ln x$ for following condition? $2\leq x <\infty$
2
votes
2answers
95 views

Proving Bounded variation is smallest linear space

Prove that $BV[a,b]$ is the smallest linear space containing all monotone functions on $[a,b].$
1
vote
1answer
48 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 ...
1
vote
1answer
41 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 ...
1
vote
1answer
149 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 ...
1
vote
1answer
56 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 $$\int_a^b |f'|\leq TV(f).$$ ...
2
votes
0answers
82 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
votes
1answer
36 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
vote
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 ...
0
votes
1answer
22 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 \, ...
1
vote
0answers
24 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
votes
0answers
35 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
vote
1answer
32 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: ...