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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1answer
472 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
276 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
176 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
88 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
82 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. ...
0
votes
1answer
274 views

Integration of a $BV$ function with respect to a finite, signed Radon measure

Let $u,w\in BV(0,1)$ be given. Since we are in dimension 1 $u$ is continuous almost everywhere and has a representation $u^{l}$ and $u^{r}$ (left, right hand side continuous). Thus we can consider the ...
2
votes
1answer
461 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
281 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
182 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 ...
1
vote
1answer
59 views

Is it true that $C_0^\ast[0,+\infty) = NBV_{loc}[0,+\infty)$

Any function from $BV_{\operatorname{loc}}[0,+\infty)$ defines a continuous linear functional on $C_0[0,+\infty)$. But is it true that any continuous linear functional on $C_0[0,+\infty)$ is given by ...
0
votes
1answer
165 views

Using Chernoff bound to analysis the Lazyselect algorithm

It's my homework of the course of randomized algorithm. In the textbook (Randomized Altorithm by Rajeev Motwani et.al.), the author analyzed this algorithm using Chebyshev bound, but are there any ...
0
votes
1answer
221 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 ...
1
vote
1answer
403 views

Uniform limit of continuous functions bounded variation

Prove or disprove that if $f:[a,b]\rightarrow\mathbb{R}$ is the uniform limit of a sequence of continuous functions each of which is of bounded variation, then $f$ is of bounded variation on $[a,b].$
0
votes
1answer
129 views

Can I make a BV function right-continuous this way?

Math people: This question is related to how can you "fix" one of the definitions of a BV function of one variable? . Suppose $f \in BV([0,1])$. I really have two-three questions. The ...
2
votes
1answer
90 views

Proving Bounded variation is smallest linear space

Prove that $BV[a,b]$ is the smallest linear space containing all monotone functions on $[a,b].$
3
votes
1answer
179 views

Math Analysis - Problem dealing with bounded variation

Let $f\colon[0,1] \rightarrow \mathbb R$ (all real numbers) be defined by $\displaystyle f(x) = x \sin \left(\frac{\pi}{2x}\right)$ if $0 \lt x \le 1$ and $f(x) = 0$ if $x=0$. Determine ...
2
votes
1answer
37 views

Inequality on function variations

Assume $f\in BV([a,b])$. Is the following true: $$(f(b)=0)\Rightarrow (V[a,b] \leqslant V[a,b) + ||f||_\infty),$$ where $VI$ denotes the total variation of $f$ on $I$.
1
vote
2answers
519 views

Continuously differentiable functions of bounded variation

From this question, we know that a continuous function of bounded variation is not necessarily absolutely continuous. But the example (Devil's staircase) given is not differentiable. What if we ...
1
vote
1answer
1k views

Some questions about functions of bounded variation: Jordan's theorem

I was trying to do some of these questions to check my understanding about the topic, but I'm not sure if they're correct. Here are my answers. 1) Suppose $f$ is continuous on $[0,1]$. Must there be ...
0
votes
1answer
114 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) ...
5
votes
1answer
74 views

questions regarding definition of bounded variation of several variables

$\int_\Omega u(x)\,\mathrm{div}\boldsymbol{\phi}(x)\mathrm{d}x = -\int_\Omega \langle\boldsymbol{\phi}, Du(x)\rangle \qquad \forall\boldsymbol{\phi}\in C_c^1(\Omega,\mathbb{R}^n)$ ...
0
votes
1answer
44 views

How to represent a bounded function

I am not completely sure whether this question belongs on mathematics.SE but I figured to give it a shot: I have a function which mathematically looks like this: $f(x)=\max(A,B\cos x)$ This will ...
2
votes
0answers
402 views

Pointwise convergence, bounded variation, and lim inf's [duplicate]

Suppose that $\{f_n\}_{n = 1}^\infty$ is a sequence of functions in $BV[0, 1]$ that converges pointwise to a function $f$ on $[a, b]$. Show that $V_a^b f \leq \liminf_{n \to \infty} V_a^b f_n$. I ...
2
votes
1answer
86 views

Construct a sequence of functions that does not converge in $B[a, b]$

Construct an example of a sequence of functions $(f_n)$ in $BV[0, 1]$ such that $f_n \to f$ uniformly on $[0, 1]$ for some function $f \in BV[0, 1]$, whereas $(f_n)$ does not converge to $f$ in the ...
3
votes
1answer
59 views

Show $BV[a, b]$ is not dense in $B[a, b]$

Show that $BV[a, b]$ is not dense in $B[a, b]$ under the metric $||f||_\infty$. I was wondering if I could get a hint.
1
vote
0answers
263 views

Solving $\int_{0}^{2\pi} \cos (x) \cos (z(x)) \, dx$ with a bounded version of $z(x)=a \cos (x)+b$

I am trying to solve the following equation: $$\int_{0}^{2 \pi } \cos (x) \cos z(x) \, dx \hspace{20 mm} (1)$$ where $z(x)$ is a bounded version of $z(x)=a \cos (x)+b$ that reads as: ...
4
votes
1answer
113 views

Total variation of a Fourier series

Let $f(x) = f(x+2\pi)$ be a bounded real function given by the Fourier series of the form $$ f(x) = \sum_{k=1}^N a_k \sin(kx + \phi_k). $$ What is the total variation $V(f)$ of this function over one ...
3
votes
1answer
191 views

Bounded Variation $+$ Intermediate Value Theorem implies Continuous

Let $I=[a,b]$ with $a<b$ and let $u:I\rightarrow\mathbb{R}$ be a function with bounded pointwise variation, i.e. $$Var_I u=\sup\{\sum_{i=1}^n|u(x_i)-u(x_{i-1})|\}<\infty$$ where the supremum is ...
2
votes
0answers
206 views

Functions of bounded variation and continuity

Suppose that $f:[a, b] \to \mathbb{R}$ is a function of bounded variation. Define $g:[a, b] \to \mathbb{R}$ by $g(x) = V_a^x f$. Show that $f$ is continuous at $x \in [a, b]$ iff $g$ is continuous at ...
1
vote
1answer
233 views

Bounded variation

For every $x \in \mathbb{R}$ define $$I(x) = \begin{cases}0 & \text{if } x \leq 0\\1 & \text{if } x > 0\end{cases}$$ Suppose that $(x_n)$ is a sequence of distinct points in (a, b) and that ...
1
vote
4answers
296 views

Bounded linear function implication

In Stephen Boyd's book boyd uses the theorem that a linear function is bounded below on $R^m$ only when it is zero. I can't really digest this. Csn someone tell me why this holds? I mean if I take a ...
9
votes
1answer
628 views

Two definitions of “Bounded Variation Function”

As far as I know, a function $f$ defined on an interval $[a, b]$ is said to be of bounded variation if $$\tag{1}V_a^b(f)=\sup\left\{\sum_{P} \lvert f(x_{j+1})-f(x_j)\rvert \ :\ P\ \text{partition of ...
1
vote
1answer
123 views

Show $\left| \int_a^b f d\alpha \right| \le \int_a^b|f|dV$ if $V$ is variation of $\alpha$ on $[a, b]$

Let $\alpha$ be of bounded variation on $[a, b]$ and assume that $V(x)$ be the total variation of $\alpha$ on $[a, x]$, $a<x\le b$ and $V(a) = 0$. Let $f$ be defined and bounded on $[a, b]$. If ...
1
vote
1answer
378 views

Why are these function of finite variation

Dealing with Itô, it simplifies a lot if you have terms which are continuous and of finite variation, since these terms have zero quadratic variation. I know that every increasing function has finite ...
0
votes
2answers
350 views

Borel Measures and Bounded Variation

What is the connection between finite Borel sign-changing measures and the functions with bounded variation on the same interval? Proof would be appreciated.
1
vote
0answers
95 views

What is Mountain Pass theorem?

The title says it all, what is mountain pass theorem, how does it work is variational problems with doubly differential constraint?
1
vote
1answer
84 views

Taylor theorem for a multivariate BV function

Given a function $f(x_1,x_2,\ldots,x_n):\Omega\to\mathbb{R}$ which is in $BV(\Omega)$ and has, in $\textbf{0}$, all partial derivatives up to order $n-1$, all equal to 0: $$ \frac{\partial^{|\alpha|} ...
2
votes
1answer
94 views

Can $f\in BV$, $f=0 a.e$ but $f'$ does not exist on an uncountable set?

I want to ask if it is possible to construct a function such that: It is of bounded variation. It is $0$ almost everywhere except on a countable set. It is not differentiable on an uncountable ...
4
votes
2answers
524 views

a problem on functions of bounded variation

Which of the following statements are necessarily true ? a. Any continuous function on [$0, 1$] is of bounded variation. b. If $f : \mathbb{R} → \mathbb{R} $ is continuously differentiable, then its ...
6
votes
2answers
2k views

Does differentiable function of bounded variation have bounded derivative?

I learned that $f$ is a function of bounded variation, when function $f$ is differentiable on $[a,b]$ and has bounded derivative $f'$. What I want to know is converse part. If $f$ is differentiable ...
5
votes
1answer
952 views

proving $f$ is absolutely continuous on $[0,1]$

I don't seem to find version of this problem in the site, but I am sure this is pretty standard type of question. $f$ be of bounded variation on $[0,1]$, and $f$ is absolutely continuous (AC) on ...
4
votes
5answers
4k views

Bounded functions have bounded derivatives.

Can the graph of a bounded function ever have an unbounded derivative? I want to know if $f$ has bounded variation then its derivative is bounded. The converse is obvious. I think the answer is "yes". ...
1
vote
2answers
312 views

is my function absolutely continuous?

If $f$ is continuous over $[a,b]$ and $\vert f\vert$ has bounded variation, is $f$ absolutely continuous? Given $\varepsilon >0$. I need to find a $\delta$ such that $\sum\vert ...
0
votes
1answer
407 views

Proving a function has a bounded variation

$f$ is continuous on $[a,b]$ and $\vert f \vert$ has a bounded variation. I would like to show $f$ has bounded variation. Using the intermediate value theorem we can take a partition such that (1) ...
1
vote
1answer
82 views

variation of $\vert f \vert^{1.5}$

$f$ has a bounded variation. $\vert f\vert ^{1.5}$ also has a bounded variation. $\vert f \vert $ is a bounded variation function, as well as integer powers of $\vert f \vert$. $\vert f \vert ^{1.5}$ ...
2
votes
0answers
553 views

Show that $f$ is of Bounded Variation by $f'$' being integrable.

For $\alpha$, $\beta$ > 0, define the function $f$ on $[0, 1]$ by $f(x) = x^{\alpha}\sin(1/x^{\beta})$ for $0 < x \leq 1$ and $f(x) = 0$ for $x=0$. Show that if $\alpha > \beta$, then $f$ is of ...
0
votes
1answer
264 views

Let $f$ be a real valued sequentially continuous function relative to a closed bounded interval $I=[a,b]$. Prove that the set $f(I)$ is bounded above

The hint that I've been given is: for each n in the naturals, use the assumption that $n$ is not an upper bound for $f(I)$ to choose a sequence of $x_n$ (from $n=1$ to infinity) in $I$; then apply ...
4
votes
1answer
454 views

An upper bound for an integral.

I want to find an upper bound of the function $h(t)$ on $t \in [0, \infty [$ with $d >1$ which is defined by $$ h(t) := \int_0^t (1+t)^d (1+t-r)^{-d} (1+r)^{-d} dr$$ and I could easily prove that ...
6
votes
1answer
784 views

Functions of bounded variation as the dual of $C([a,b])$

I am trying to understand this proposition about the dual of $C([a,b])$. I would like some help with the following: (1) What does the integral with respect to a function of bounded variation mean? ...
2
votes
1answer
176 views

$f(0)=0,\;\;f(x)=e^{-2/x}\sin\left(e^{1/x}\right)$, is $f$ bounded variation on [0,1]?

Let $$f(0)=0,\;\;f(x)=e^{-2/x}\sin\left(e^{1/x}\right),$$ is $f$ bounded variation on $[0,1]$? Here is my thinking: Since $f$ is differentiable on $(0,1]$ and continuous on $[0,1]$ If $f^\prime$ ...