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
54 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 ...
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1answer
18 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
8 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 ...
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1answer
20 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 ...
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1answer
119 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 - ...
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1answer
23 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 ...
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1answer
28 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 ...
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1answer
53 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 ...
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1answer
57 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 ...
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2answers
27 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
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1answer
100 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 ...
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0answers
63 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 ...
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1answer
46 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 ...
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1answer
28 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 $g$, ...
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1answer
174 views

Inequality on Functions of Bounded Variations

The following discussions were based on Schwabik's book, entitled "Generalized Ordinary Differential Equations", pp.26-29. Let $\delta$ be a positive function defined on $[a,b]$. We say that ...
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0answers
86 views

How to prove $\int_{a}^{b} f dĪ±$ = $\int_{a}^{b} f dĪ²$

I'm working on an exercise from Carothers' Real Analysis: where $BV[a,b]$ denotes the set of all functions on [a,b] with bounded variation and $C[a,b]$ is the set of all continuous functions on ...
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0answers
50 views

Absolute continuity implies bounded variation

Let $f$ be absolutely continuous on $[a,b]$. I want to prove that $f$ is of bounded variation. I am reading Royden and Fitzpatrick so they use the following notations: Let ...
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1answer
38 views

A special approximation of BV functions

Suppose $\Omega:=B(0,1)\subset \mathbb R^2$. Let $u\in BV(\Omega)$ be a radially symmetric, i.e., $u(x)=u(Rx)$ for all $R\in SO(2)$. In addition, suppose $w$ to be an affine function,. i.e., ...
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0answers
34 views

p-variation of semimartingales

Does every (particularly continuous) semi-martingale have bounded 2+$\epsilon$-variation for all $\epsilon>0$? Note that I am not asking, whether they have finite quadratic variation - that is ...
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2answers
66 views

Multiplication of absolutely continuous function and function of bounded variation

Is the product of an absolutely continuous function $f$ and a continuous function of bounded variation $g$ on $[0,1]$ for which $f(0)=0$ and $g(0)=0$, absolutely continuous?
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1answer
100 views

If a derivative $f^\prime$ is of bounded variation then prove that $f^\prime$ is continuous.

I am having trouble with the following problem: Let $f:[a,b]\to \mathbb R$ be differentiable on $[a,b]$ and $f^\prime$ is of bounded variation on $[a,b]$. Prove that $f^\prime$ is continuous ...
3
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1answer
49 views

Show that $\,f(x) = \sum_{n=0}^\infty a_nx^n$, for $x \in [0,1]$, is of bounded variation

Let $\{a_n\} \subset \mathbb{R}$, be such that $\sum_{n=0}^\infty \lvert a_n\rvert < \infty$. Define $$f(x) = \sum_{n=0}^\infty a_nx^n \quad \text{for } x \in [0,1]$$ Prove that $f$ is of bounded ...
1
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1answer
36 views

Boundedness of a modular form in $\mathbb{H}$

Let be $k>0$ and $f \in S_k(\Gamma)$. I want to show that the function $h(z)=Im(z)^{\frac{k}{2}}\cdot |f(z)|, \; z\in\mathbb{H}$ is bounded in $\mathbb{H}$. I have already shown, that $h$ is ...
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1answer
16 views

Semimartingale jumps question

I am reading a statement which contains $\Delta X \cdot Y$ where $X$ is a semimartingale and $Y$ is a finite variation process and the notation means the lebesgue stieltjes integral. My problem is ...
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2answers
77 views

Finding upper and lower bounds on a trigonometric function

I've been tasked with finding the upper and lower bounds of the element: $A = sin(\frac{\pi.n}{2n+3}) | n\in\mathbb{N}$ I think I have found the upper bound by doing: $\lim_{n\to +\infty} ...
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1answer
44 views

Why is this sequence relatively compact in $L^1$?

I am currently reading this paper from 1973. In short, one has given a linear continuous operator $P : L^1([0,1]) \to L^1([0,1])$ with ||P||=1 and for $f \in L^1$ a family of functions ...
0
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0answers
42 views

Finite Variation Function.

Let $V$ be a right continuous BV function and put $V_t = \int_0^t a_s dC_s$ where $C$ is increasing and right continuous. Is it true that if $V$ is continuous then $\int_0^t |f_s a_s| dC_s < ...
0
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1answer
30 views

Prove convergence of this sequence $f(n)_{n \in \mathbb{N}}= \left(\frac{10+in}{n^2 + 2in}\right)^n$

I am having this sequence $f(n)_{n \in \mathbb{N}}= (\frac{10+in}{n^2 + 2in})^n$ Is this sequence bounded/ convergent? Thoughts: $lim_{n \to \infty}(\frac{10+in}{n^2 + 2in})=lim_{n \to ...
0
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1answer
63 views

Find $f \in B[0,2] $ and $g$ Riemann-Stieltjes integrable on $[0,1]$ and $[1,2]$ but not in $[0,2]$

I'm stuck at one exercise: Find $f \in BV[0,2]$ and $g$ such that $g$ is Riemann-Stieltjes integrable in $[0,1]$ and $[1,2]$, that is, $$\int_{0}^1 \ g \ df$$ and $$\int_{1}^2 g \ df$$ exist, but ...
0
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1answer
32 views

Alternate definition on bounded variation

If $g:[a,b]\to \mathbb{R}$ the $g$ is of bounded variation iff $$TV(g,[a,b])=\sup\sum_{i=1}^n |g(t_i)-g(t_{i-1})|<\infty$$ where the supremum is taken over all partitions ...
0
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1answer
63 views

Showing that $f$ is not Absolutely continuous

Frist:- I am not sure about what title this question should be. Suppose the function $f:[0,\frac{1}{2}]\rightarrow \mathbb{R}$ defined by $$ f(x) = \begin{cases} 0, & \text{if }x=0 \\ x ...
2
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0answers
108 views

The Co-area formula for $BV$ function V.S. the co-area formula for $C^\infty$ functions

I am working on the proof of the co-area formula for $BV$ functions. Suppose $u\in BV(\Omega)$ then the co-area formula states that $$ \|Du\|(\Omega)=\int_{-\infty}^\infty \|\partial ...
13
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4answers
224 views

A continuous function on $[0,1]$ not of bounded variation

I'm looking for a continuous function $f$ defined on the compact interval $[0,1]$ which is not of bounded variation. I think such function might exist. Any idea? Of course the function $f$ such ...
4
votes
1answer
102 views

Bounded variation problem

Let $x=0.a_1a_2\dots$ be the decimal expansion of a number $x$, $0<x<1$. If two decimal expansions of $x$ exist, the one that ends with $0$ā€™s is taken. For what values of $q > 1$ is the ...
2
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1answer
82 views

Calculating explicitly the total variation of $x^2 \sin\left(\frac{\pi}{2x}\right)$ on $[0,1]$.

I am attempting to calculate the total variation of the function $f$ on [0,1] defined as $$f(x) = \begin{cases} x^2 \sin\left(\frac{\pi}{2x}\right) \text{ if } x\neq 0 \\ 0 \qquad \qquad \text{ if } ...
0
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1answer
83 views

Is $x^{3/2}\sin(\frac{1}{x})$ of bounded variation?

I'm trying to show that $f(x)$ is of bounded variation where $f(x)=x^{3/2}\sin(\frac{1}{x})$ on $[0,1]$. I think that it is but I can't show it explicitly. Any help will be appreciated.
1
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1answer
37 views

Bounded variation of function $(f-M)^+$ and the measure of the set where it is concentrated

I read this statement in the book by Evans & Gariepy, page 215, last two lines. Here $f\in BV(R^n)$ and for fixed $\epsilon>0$ and $N>0$, we define $$A_\epsilon^N :=\{x\in \mathbb ...
2
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1answer
42 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\{ ...
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1answer
56 views

Total variation of a sequence of functions

I have this problem, Let $\{f_n\}$ be a sequence of real valued functions on $[a,b]$ that converges at each point of $[a,b]$ to a function $f$. Then $T_a^b(f)ā‰¤\liminf T_a^b(f_n)$. It is solved in, ...
2
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1answer
105 views

The derivative of total variation

Define function $u$ on $[0,1]$ such that $$ u(x)= \begin{cases} x^2\cos\frac{1}{x} &0<x\leq 1\\ 0 & x=0 \end{cases} $$ and by $V(x):= \text{Var}_{[0,x]}u$, i.e., the total variation of ...
1
vote
1answer
32 views

The behavior of BV functions at a point of approximate continuity

Given $u\in BV(\mathbb R^N)$, we say $u$ is approximate continuos at $x$ and the approximation limit is $l\in R$ if $$ \lim_{r\to 0}\frac{\mathcal{L}^N(B(x,r)\cap \{|u-l|>\epsilon\})}{r^N} =0 $$ ...
1
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0answers
29 views

Bound of a fourier series when coefficients are bounded

Let $f(x)$ be a finite fourier series with $$f(x)=a_0+\sum_{n=1}^N\left(a_n\sin{\left(2\pi nx/P\right)}+b_n\cos{\left(2\pi nx/P\right)}\right)$$ and bounded coefficients ...
0
votes
1answer
26 views

If $\alpha,\beta \in BV[a,b]$, prove that $|a| \in BV[a,b]$ and $\min(\alpha,\beta),\max(\alpha,\beta) \in BV[a,b]$.

Question: If $\alpha,\beta \in BV[a,b]$, prove that $|a| \in BV[a,b]$ and $\min(\alpha,\beta),\max(\alpha,\beta) \in BV[a,b]$. Attempt: Since $\alpha \in BV[a,b]$, $$ \sum_{i=1}^n \left| ...
0
votes
1answer
88 views

Lebesgue decomposition of an increasing function

This problem asserts that any increasing function $f$ on $[a,b]$ can be decomposed as $$F=F_A+F_C+F_J,$$ with $F_A$ is absolutely continuous, $F_J$ is a jump function, and $F_C$ is a singular ...
0
votes
1answer
27 views

The algebraical properties of $BV(\Omega)$

We know in one dimension that $BV(I)$ is an algebra, for both $I$ is bounded or unbounded. Now suppose $\Omega\subset R^N$ is open bounded with Lipschitz boundary. My question is: is it $BV(\Omega)$ ...
2
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1answer
29 views

Question about finite preimeter in $BV$ space

Given $\Omega\subset R^N$ is open bounded, we say $E\subset \Omega$ has finite perimeter in $\Omega$ if $\chi_E\in BV(\Omega)$. Follows from Evans & Gapriep's book, we write for any $\varphi\in ...
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0answers
36 views

Counterexample for Trace operator in BV space.

Suppose $\Omega\subset R^N$ is open bounded with Lipschitz boundary. Then for $u\in W^{1,p}(\Omega)$, there exists a linear operator $T$: $W^{1,p}(\Omega)\to L^{p}(\partial\Omega)$ such that the ...
-1
votes
1answer
32 views

Find $K$ such that $|(x, y)| > K$ implies $(x - 1)^2 + (y + 2)^2 > C+ 4$.

For any š¶ āˆˆ ā„, find š¾ such that |(š‘„, š‘¦)| > š¾ ā‡’ š‘„2 + š‘¦2 - 2š‘„ + 4š‘¦ + 1 > š¶ i.e. (š‘„ - 1)Ā² + (š‘¦ + 2)Ā² > š¶ + 4 whenever |(š‘„, š‘¦)| > š¾ NOTE: š¾ is a function of š¶ only, and does NOT depend ...
0
votes
2answers
84 views

Which of the following is true about $f(x)$?

If $f(x)=x+\sin x$, then which of the following is true about $f(x)$? $1.f(x)$ is uniformly continuous on $\mathbb{R}$. $2.f(x)$ has bounded variation on $\mathbb{R}$. $3.f(x)$ does not have ...
0
votes
1answer
24 views

First Order Variation Example

I am in a introductory stochastic calculus class and came across and example from that asks for the first order variation and the quadratic variation of a continuous function. For example: $f(x) = ...