2
votes
1answer
62 views

Unbounded variation but differentiable everywhere

A function with bounded variation is differentiable almost everywhere. There are also functions with unbounded variation that are differentiable almost everywhere (e.g. take ...
1
vote
0answers
30 views

Question on Bounded Variation involving partitions

If $f\colon[a,b]\rightarrow \mathbb{R}$ and $P=(a=x_0, x_1, ...,x_n=b)$ is any partition of $[a,b]$ let $v_f(P)$ be defined as $$v_f(P) = \sum_{k=1}^n |f(x_k) - f(x_{k-1})| $$ and the total variation ...
1
vote
1answer
26 views

Question on Functions of Bounded Variation

We say that $f$ is of bounded variation over $[a,b]$ or $f\in BV[a,b]$ if $V_f[a,b] = \sup \sum_{k=1}^n |f(x_k) - f(x_{k-1})| < \infty, $ where the supremum is taken over all possible partitions ...
1
vote
1answer
53 views

Does bounded and continuous implies Lipschitz?

If a function $f : \mathbb{R} \rightarrow \mathbb{R}$ is integrable, bounded and continuous, is it also Lipschitz continuous?
1
vote
0answers
36 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. ...
3
votes
0answers
37 views

Understanding the duals of $L^{\infty}(K)$ and $C(K)$ and weak-* compactness

Let $(K\subset \mathbb{R}^n,\mathcal{B}(K),\lambda)$ be a measure space, where $\lambda$ is the Lebesgue measure and $K$ is compact. According to Wikipedia (with adapted notation), The dual ...
2
votes
2answers
28 views

Continuity and Inverse Preservation of Boundedness Implies Preservation of Closed Sets

Sorry about the rather long title - wasn't sure what else to call it! Here is my question: Let $f : \Bbb R^n \rightarrow \Bbb R^m$ be continuous and such that $f^{-1}(F)$ is bounded whenever $F$ ...
1
vote
1answer
38 views

Does bounded variation imply boundedness

Using the standard definition $$||f||_{TV} := \sup_{x_0<\cdots<x_n}\sum_{i=1}^{n} |f(x_i) - f(x_{i-1})|.$$ 1.When the domain is a bounded interval $[a,b]$, the statement holds. 2.When the ...
3
votes
1answer
126 views

Function coincides with a function of bounded variation almost everywhere

Problem Suppose $f\in L^1(\mathbb R)$ satisfies that there exists $A\ge0$ such that $$\int_{\mathbb R}\lvert f(x+h)-f(x)\rvert dx\le A\lvert h\rvert$$ for all $h\in\mathbb R$. We need to show that ...
0
votes
1answer
25 views

A question related to bounded variation

Let $f\in C^{BV}([0,1])$ (i.e. continuous and has bounded variation). Let the intervals $I$ and $T$ satisfy the following: $I\subset T\subset [0,1]$ and for sufficient small $\delta>0$, ...
3
votes
1answer
52 views

Definition of total variation

According to wikipedia, the total variation of the real-valued function $f$, defined on an interval $[a,b]\subset \mathbb{R}$, is the quantity $$V_b^a=\sup_{P\in\mathcal{P}}\sum_{i=0}^{n_P-1}\left | ...
3
votes
1answer
58 views

Differentiable function in n dimensions

If a function $f:\mathbb R^{n} \rightarrow \mathbb R^m $ is differentiable at a point $a$ can we say that there is a neighbourhood of $a$ such that $f$ is locally Lipschitz? (i.e. there is some ...
0
votes
1answer
59 views

Bounded variation

Define $$g(x) = \begin{cases} x^2\cos\left(\frac{1}{x}\right) &\mbox{if } x \neq 0\\ 0 & \mbox{if } x = 0. \end{cases} $$ Is g of bounded variation on $[-1,1]$. My attempt:-To show that $g$ ...
0
votes
1answer
40 views

Problem on Bounded Variation

Assume $f$ is of bounded variation on $[a,b]$. Show that there is a sequence of partitions $\{P_n\}$ of $[a,b]$ for which the sequence $\{TV(f,P_n)\}$is increasing and converges to $TV(f)$? ...
1
vote
2answers
38 views

Need guidance on a problem about oscillation from Spivak's Calculus on Manifolds

I've been stuck on this particular problem for a while now: Let $f: [a,b] \rightarrow \mathbb{R} $ be an increasing function. If $x_1, ... ,x_n \in [a,b]$ are distinct, show that $\sum\limits_{i=1}^n ...
2
votes
0answers
62 views

Arzela Ascoli variation theorem

I am trying to prove the following variation of the Arzel-Ascoli theorem: Let $(f_n)n$ be a sequence of differentiable functions on $[a,b]$ whose derivatives are uniformly bounded and there is an ...
0
votes
1answer
20 views

Definition of Bounded Variation Function with vectorial arguments

I try to found a definition of a function $$f(x)\colon\mathbb R^m\to \mathbb R^n$$ that use the norm. Is the formula below correct? $$TV=\sup\sum_{i=1}^k \|f(x_i)-f(x_{i-1})\|.$$ with k any finite ...
2
votes
1answer
40 views

Monotone convergence for monotone functions in BV

For $n \geq 1$ let $f, f_n : [0, 1] \to \mathbb{R}$ be monotone nonincreasing functions. Suppose that $f_n \nearrow f$ pointwise monotonically as $n \to \infty$. Is then $\mathrm{TV}(f - f_n) \to 0$ ...
3
votes
3answers
1k views

If a function $f(x)$ is Riemann integrable on $[a,b]$, is $f(x)$ bounded on $[a,b]$?

Most statements regarding Riemann integrals (at least the ones that I have encountered) begin with the statement "for $f(x)$ bounded on $[a,b]$." I am wondering if Riemann integrability implies ...
0
votes
1answer
77 views

Finding the total variation of $3x^2-2x^3$ [closed]

I would appreciate if someone could help me to find the total variation of $3x^2-2x^3$ on $[-2,2]$. Thanks
0
votes
1answer
270 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 ...
3
votes
2answers
371 views

If f is of bounded variation is f Riemann integrable?

I want to know if f is of bounded variation on [a,b] does it follow that f is Riemann integrable on [a,b]?
0
votes
1answer
19 views

A problem that involves a supremum

We let $[a,b]\subset \mathbb{R}$ and $f:[a,b]\to \mathbb{R}$. How do we prove that $$\sup \left|\sum_{i=1}^n a_i[f(x_i)-f(x_{i-1})]\right|=\sup\sum_{i=1}^n|f(x_i)-f(x_{i-1})|$$ where $a_i\in [-1,1]$ ...
3
votes
1answer
87 views

Continuity of bounded variation functions

A function $f:\mathbb{R}\rightarrow\mathbb{R}$ is a BV function if there exists $M<\infty$ for which $$\sum_{k=1}^N|f(x_k)-f(x_{k-1})|\leq M$$ for every sequence $x_0<x_1<\ldots<x_N$ and ...
0
votes
2answers
48 views

Proving inequality regarding arc length in Apostol's book

If P={$t_0,t_1,...,t_m$} is a partition of [a,b] prove that the following inequality holds. ...
1
vote
2answers
281 views

Is there an unbounded function with a bounded derivative?

I know that there exists bounded functions with unbounded derivatives. For example, $\sin(e^x)$ is bounded and differentiable everywhere on $\mathbb{R}$, but its derivative is unbounded. Is it ...
0
votes
1answer
34 views

On functions that are bounded by other certain functions

I am trying to address a specific, but also rather abstract, problem, which briefly can be stated as follows: Let $f$, $g$ be real-valued functions defined for all column vectors in $\Re^n$, i.e., ...
0
votes
1answer
67 views

Proving that total variation is equal to $\int_{a}^{x}|g|$

Question:Suppose $g$ is continuous on $[a,b]$. Let f(x)=$\int_{a}^{x}g$ where $x∊[a,b]$. Show that $\int_{a}^{x}|g|$ gives the total variation of $f$ on $[a,x]$. I managed to prove that ...
0
votes
1answer
49 views

A way to find the total variation of a polynomial if the zeroes of the derivative are known.

I came across a question in Apostol's book which said to describe a method to find the total variation of a polynomial if the zeroes of the derivative of it is known ( points at which the derivative ...
1
vote
0answers
98 views

Every polynomial bounded on [a,b] is of bounded variation.

Question: Prove that every polynomial bounded on the compact interval [a,b] is of bounded variationMy proof: Let $f(x)$ be a bounded polynomial of n degree. Then $f '(x)$ is also a polynomial bounded ...
0
votes
1answer
226 views

Integration of functions with bounded variation

I need to prove that if a function $f: [a,b] \to \mathbb{R}$ has bounded variation, than $f$ is integrable on $[a,b]$. This is what I have tried: Let $S(P)$ and $s(P)$ denote the upper and lower ...
1
vote
2answers
27 views

Greatest Lower Bounded Irrational?

If all elements of $S$ are irrational and bounded from below by $\sqrt 2$ then $\inf S$ must be irrational . I would say this statement is true since $S=\{ \sqrt 2, \sqrt 3, \sqrt 5,\ldots\}$ the ...
1
vote
4answers
121 views

Does extreme value theorem hold when continuous is replaced with bounded?

The extreme value theorem says that if the domain of a 'continuous' function is compact then both the max and min of the function lies in the domain set. My question is: can the 'continuity' be ...
0
votes
0answers
43 views

The Leibniz's rule for bounded variation functions

Let M be a Riemannian manifold, U is a bounded domain in M. If f,g are bounded variation function on U, then $$(fg)_{x_i}=f_{x_i} \cdot g +g_{x_i} \cdot f$$? How to prove? And for the mollified ...
1
vote
2answers
62 views

Examples of $f \in C^2[a,b]$ where the total variation of $f$ on $[a,x]$ is not in $C^2[a,b]$

Suppose $f:[a,b]\to\mathbb{R}$ is of bounded variation; define $V(x) = V[f;a,x]$ (the total variation of $f$ on $[a,x]$. I want to show that $V \in C^1[a,b]$. Since $f'$ is continuous, hence bounded ...
1
vote
1answer
196 views

Absolutely continuous functions and general absolute continuity

First, the definitions: $f$ is AC on $E$ if $$\forall \epsilon >0\ \exists \delta >0\ \forall \{[a_k,b_k]\}_{k=1}^N \mbox{ such that }a_k,b_k \in E,\ \Sigma(b_k - a_k) <\delta : \Sigma| ...
2
votes
1answer
128 views

How to show that every bounded variation function on $[a,b]$ is differentiable a.e?

I'm struggling with this question for over a week now. I know the proposition is true, but haven't managed to prove it yet. any suggestions anyone? ($f$ is BV on $I$ if ...
2
votes
1answer
154 views

Total Variation and indefinite integrals

Suppose $f$ is Lebesgue integrable on $[a,b]$ and $F(x) = \int^x_a f(t) dt$, $x \in [a,b]$. Show that $F$ has bounded variation, and the total variation $T^b_a(F)$ satisfies $$ T^b_a(F) = \int^b_a ...
1
vote
1answer
102 views

About Regulated and Bounded Variations in Banach Spaces

In the following definitions, we assumed that $(X,\left\|\cdot\right\|)$ is a Banach space. Definition 1. $f:[a,b]\to X$ is of bounded variation on $[a,b]$ if ...
1
vote
1answer
62 views

if $f_n\to f$ uniformly in $[a,b]$ then $f\in BV$

While preparing to test in calculus I found the question above: Let $f_n(x)\to f$ uniformlly on $[a,b]$. Prove or give counterexample: if $\forall n\in\mathbb N f_n(x)\in BV$ and $\exists M>0$ ...
1
vote
1answer
262 views

Two questions on Lebesgue Decomposition of an increasing function?

I come up with the question in doing Stein's Real analysis, Chap3. Ex. 24, which assert that any increasing function $f$ on $[a,b]$ can be decomposed as $$F=F_A+F_C+F_J,$$ with $F_A$ is absolutely ...
3
votes
1answer
321 views

Bounded variation and $\int_a^b |F'(x)|dx=T_F([a,b])$ implies absolutely continuous

If $F$ is of bounded variation defined on $[a,b]$, and $F$ satisfies $$\int_{a}^b |F'(x)|dx=T_F([a,b])$$ where $T_F([a,b])$ is the total variation. How to prove that $F$ is absolutely continuous? ...
1
vote
1answer
168 views

Which are functions of bounded variations?

Let $f, g : [0, 1] \to \mathbb{R}$ be defined as follows: $f(x) = x^2 \sin (1/x)$ if $x = 0$, $f(0)=0$ $g(x) = \sqrt{x} \sin (1/x)$ if $x = 0, g(0) = 0$. Which are functions of bounded ...
2
votes
1answer
114 views

Bounded Variation is Bounded

Let $f:[a,b]\to \mathbb{R}$ and let $D=\{x_o,x_1,...,x_n\}$ be a division of $[a,b]$. We say that $f$ is of bounded variation on $[a,b]$ if $\displaystyle \sup_{D\in \mathscr{D}} \sum_{i=1}^n ...
0
votes
1answer
67 views

A clarification about BV functions.

From definition, a locally integrable $ u \in BV(\Omega) $ if its distribution derivative is given by a signed Radon measure. That is there exists $ \mu $ such that for any $ \phi \in ...
2
votes
2answers
75 views

Prove that this function has finite variation

We know that $f$ has finite variation on $[a,b]$. Prove that $$g(x)= \begin{cases} 0, & x=a\\[8pt] \frac{1}{x-a} \int _{a} ^x f(t) \, dt , & x \in (a,b] \end{cases} $$ has finite variation. ...
1
vote
1answer
403 views

Taking the derivative of an integral of a discontinuous function

When I took measure theory with Frank Jones' books years ago, I did every problem in the book because I loved its teaching style. There was one problem that took me 4-5 years to solve. It was problem ...
3
votes
1answer
184 views

Bounded variation implies Borel measurable

Suppose that $f\colon[a, b] \to \mathbb{R}$ is a function of bounded variation. Show that $f$ is Borel measurable. I was wondering if I could get a hint.
4
votes
2answers
160 views

Find f ae-differentiable with $f´\in L^1(0,1)$ but not in $BV$…

Here is a natural question which I didn't find in Measure Theory books: Construct a continuous function $f(x)$ in $[0,1]$ with derivative at ae $x\in(0,1)$, and so that $f'(x)\in L^1(0,1)$, but such ...
0
votes
1answer
161 views

which condition says that $f$ is necessarily bounded variation

Which of the following condition below imply that the $f:[0,1]\to\mathbb{R}$ is necessarily Bounded Variation? monotone; continuous and monotone; has derivative on $(0,1)$; bounded derivative on ...