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
127 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
34 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
284 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
619 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
103 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
57 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
41 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 ...
1
vote
1answer
247 views

Pointwise convergence, bounded variation, and lim inf's

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
83 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
55 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
243 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
102 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
136 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
152 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
169 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
177 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 ...
0
votes
1answer
80 views

Some questions on measure and BV

If $S\subseteq \Bbb R$ and $m(S) >0$, then show that for all $c \in (0,1)$ $\exists I$ such that $ {m(S\cap I)} > c\cdot m(I)$ If $f \in BV[a,b]$ and $f\ge 0$, then show $2^f$, $\sqrt{f}$, ...
8
votes
1answer
351 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
104 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
206 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
248 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
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0answers
75 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
72 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
80 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
344 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
929 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
763 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
2k 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
243 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
266 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
79 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
447 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
186 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
307 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
558 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
2answers
163 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 $f^\prime$ ...
1
vote
1answer
66 views

Intuition for Rough path

I pray to kindly show the intuition behind the concept of rough path. Google provided some links that deal with the notion of rough path but was difficult for me to have an idea.
2
votes
4answers
202 views

$\{z\in C:|z| = |\operatorname{re}(z)| +|\operatorname{im}(z)|\}$ open or closed [duplicate]

Possible Duplicate: Proving that a complex set in open/closed/neither and bounded/not bounded I think $\{z\in C:|z| = |\operatorname{re}(z)| +|\operatorname{im}(z)|\}$ is closed. But I have ...
0
votes
1answer
100 views

Functions of bounded variation on $\Bbb R$

How can one define the total variation of a function of bounded variation on $\Bbb R$? i.e., how one can evaluate the total variation on infinite intervals?!
3
votes
1answer
175 views

Functions of bounded variation and $L^\infty$ functions

Here are my questions. Are $L^\infty$ functions of bounded variation? Is the composition of two BV functions still of bounded variation? Is $x\mapsto \frac 1{f(x)}$ of bounded variation when $f$ ...
4
votes
2answers
583 views

upper bound of exponential function

I am looking for a tight upper bound of exponential function (or sum of exponential functions): $e^x<f(x)$ when $x<0$ or $\sum_{i=1}^n e^{x_i} < g(x_1,...,x_n)$ when $x_i<0$ Thanks a ...
0
votes
2answers
214 views

Limit Points within a Set

If I have an uncountable subset $A \in \mathbb{R}$, and we assume A is nonempty, does it follow that every point within $A$ is a limit point of $A$ from the density of $\mathbb{Q}$ in $\mathbb{R}$ ...
2
votes
4answers
278 views

a tetration limit for base $a > e^{1/e}$

Let $a$ be a real number with $a > e^{1/e}$ and $a <> e$. $slog$ means superlog base $e$ and $sexp$ means superexp base $e$. $sloga$ means superlog with base $a$ and $sexpa$ means superexp ...
-4
votes
1answer
117 views

Boundedness on strips in the complex plane for functional equations [closed]

We know that the recurrence for $b>0$ (1) $f(0)=1$ (2) $f(z+1)=b{f(z)}$ has $f(z)=b^z$ as the only entire solution that is bounded on the strip $S=\{z: 0<\Re(z)\le 1\}$. The image of $S$ ...
5
votes
1answer
284 views

Absolute continuity

$f$ is continuous and of bounded variation on $[0,1]$, $f$ is absolutely continuous on any $[c,1]$ with $c \in (0,1]$. Then $f$ is absolutely continuous on $[0,1]$. How to show this? ...
2
votes
0answers
166 views

Jordan decomposition measure and its distribution function

Let $f: \mathbb{R} \rightarrow \mathbb{R}$, and for $x \in \mathbb{R}$, let $T_f(x)=\sup \left\{\sum_{j=1}^N \left|f(x_j)-f(x_{j-1})\right|\right\}$, where the supremum is taken over all $N \in ...
2
votes
1answer
206 views

Fourier transform of derivative for the bounded function

I was going through the derivation of Wigner distribution properties, and encountered a certain step in the proof I could not justify. Namely, the step requires the following equality to be true: ...
1
vote
0answers
109 views

Non-periodic BV function

I want to know the definition of non-periodic bounded variation function. I know the definition for periodic function of bounded variation, which is, Let $f:[a,b]\to \mathcal c$ and ...
4
votes
3answers
994 views

If a function $f$ is differentiable on $[a,b]$ and its derivative $f^\prime$ is integrable, must $f$ be of bounded variation?

I know there is a theorem saying if $f$ defined on $[a,b]$ is of bounded variation, then it is differentiable on $(a,b)$ a.e and $f'$ is integrable over $[a,b]$. I wonder whether the converse is ...
6
votes
1answer
157 views

Is this equivalent to bounded variation?

Let $\Omega \subset \mathbb{R}^n$ be open and bounded with smooth boundary. For $f \in L^1(\Omega)$, define $$ \|D_1 f\|_M(\Omega) =\inf\left\{\liminf_{k\to\infty}\int_\Omega |\nabla f_k|\,dx \mid ...