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
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
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1answer
160 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
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1answer
215 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
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1answer
389 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 $[a,b].$
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1answer
127 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
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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 $[a,b].$
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0answers
21 views

Positive solutions for third order semipositone boundary value problems

please what is "semipositone" http://www.sciencedirect.com/science/article/pii/S0893965909000500 thak you
3
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1answer
168 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
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1answer
36 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 $I$.
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2answers
495 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
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1answer
987 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
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1answer
113 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
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1answer
69 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
1answer
387 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
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1answer
85 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
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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.
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0answers
259 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
187 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
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0answers
197 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
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1answer
226 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
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4answers
281 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 ...
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1answer
575 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
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1answer
118 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
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1answer
336 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
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2answers
340 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
91 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?
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1answer
79 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
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1answer
92 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
501 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
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2answers
1k 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
936 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
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5answers
3k 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
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2answers
308 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
387 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
549 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
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1answer
262 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
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1answer
433 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
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1answer
750 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
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1answer
174 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 $f^\prime$ ...
2
votes
1answer
90 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.
3
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4answers
212 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
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1answer
115 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
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1answer
240 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
1k 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
284 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
294 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 ...
5
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1answer
327 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? ...