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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2
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1answer
187 views

Integrating exponential of multiple exponentials

I have a integral term that looks similar to $\int_0^\infty\exp(-u-ae^{-c_1u}-be^{-c_2u})\,du$ where the constants $a,b,c_1,c_2>0$. For the case where $b=0$ I can use the answer from: Integrating ...
1
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0answers
97 views

How to show that the first derivative is bounded in a function

\begin{equation} y=\arccos\left( -\frac{1}{2\left(Dr^{\dfrac {|\sin(2x+\theta)|}{M\sin x\sqrt{A+2B\cos(2x+\theta)}}}+1\right)} \right) \nonumber \end{equation} How to show in above function the ...
1
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3answers
696 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
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2answers
34 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 ...
2
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1answer
778 views

is uniform convergent sequence leads to bounded function?

Suppose there is there is uniform convergent sequence $(f_n)$ on the set $A$, and each $f_n$ is bounded on $A$, i.e., there exist $M_n>0$ such that $|f_n(x)|\le M_n$ for all $x\in A$ Is it true ...
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1answer
65 views

Prove that $\log_2 n$ is not bounded polynomially from below, need 2nd step

i.e. that $\log_2 n\not\in\Theta(n^x)$ for any $x > 0$ i shall not use induction on $x$ ( as $x = 1$ base case etc) my guess is : i use the def. of big theta: $$ 0≤c_1·n^x \le \log_2 n \le c_2· ...
2
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4answers
227 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 ...
3
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4answers
216 views

Stability theory: Every solution of the scalar equation: $\ddot{x}+\left [a+b(t) \right ]x=0$ is bounded in $\left [t_0, +\infty \right )$.

EDITED Prove that: If $a>0$ and $$\int_{t_0}^{\infty} |b(t_1)|\mathrm{d}t_1<+\infty$$ then every solution of the scalar equation: $$\ddot{x}+\left [a+b(t) \right ]x=0$$ is bounded in $\left ...
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2answers
77 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 ...
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2answers
301 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| ...
0
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1answer
66 views

Coercive problems

this is the complete problem and i have a problem that is : i dont understand step 2: step 1:"shows that $m>-\infty$ i dont understand how to prove it ? can someone help me please ? thank ...
0
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2answers
55 views

About tight rational bounds

Suppose we have rational functions $f$, $g$, and $h$ defined for all natural numbers $n$ such that $f \leq g \leq h$ for all $n \in \mathbb{N}$. How can we prove that there is no rational functions ...
2
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1answer
186 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 ...
3
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0answers
101 views

Discontinuous local martingales with finite variations

I would like to know if a discontinuous local martingale with paths of finite variations almost surely is a martingale. I feel that it should be the case but can't find a straightforward argument. As ...
4
votes
1answer
350 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
54 views

Does the squared root $\sqrt{|\cdot|}$ belong to $BV((-1,1))$?

Does the squared root belong $\sqrt{|\cdot|}$ to $BV((-1,1))$? In an affirmative case, what is its derivative in the distributional sense, i.e., what is the Radon measure $\mu$ such that $\mu=Du$ ?
2
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1answer
207 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
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1answer
41 views

Strict inequality in lower semicontinuity of BV functions

Does anybody know an example of a sequence $f_k \in BV(\mathbb{R^n}) \ $ where $n>1$ and $f \in L^{1}_{\operatorname{loc}}(\mathbb{R^n})$ such that $ f_k \rightarrow f \ $ in ...
1
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1answer
63 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 $\exists M>0$ ...
2
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1answer
266 views

About functions of bounded variation

I got the following the following idea in one of the articles that I'm reading. It goes this way. Let $X$ be a Hausdorff topological vector space and let $\mathcal{D}$ be the family of all divisions ...
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3answers
3k views

Upper Bound of Logarithm

For $1\leq x < \infty$, we know $\ln x$ can be bounded as following: $\ln x \leq \frac{x-1}{\sqrt{x}}$. Then what is the upper bound of $\ln x$ for following condition? $2\leq x <\infty$
1
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1answer
640 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 ...
2
votes
1answer
142 views

Integration with respect to a non-decreasing function on $\mathbb{R}$

Let $\alpha(t)$ be a non-decreasing function on $\mathbb{B}$ and consider the integral $$ \int_{-\infty}^{+\infty} e^{-xt}d\alpha(t) $$ absolutely convergent in $I$. Does exist a measure ...
3
votes
1answer
589 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
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1answer
359 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
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1answer
234 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
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1answer
102 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
87 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
344 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
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1answer
487 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 ...
4
votes
1answer
356 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
186 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
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1answer
64 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
215 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
245 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 ...
2
votes
1answer
514 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
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1answer
139 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
2answers
95 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
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1answer
206 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
41 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
655 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 ...
2
votes
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
122 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
86 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
51 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
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0answers
404 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
88 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
60 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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vote
0answers
274 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
138 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 ...