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
775 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 ...
1
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1answer
73 views

A question on a function of bounded semivariation

Let $(X,\tau)$ be a Hausdorff locally convex TVS and let $P(X)$ be a family of continuous seminorms on $X$ that generates the topology $\tau$. I got the following definition from one of the articles ...
3
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2answers
897 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
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1answer
26 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 $a_i\in [-1,1]$ ...
0
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1answer
52 views

Function of bounded variation and cardinal of set of discontinuities.

Let $f:[a,b] \to \mathbb R$ such that the image is a finite set. Prove that $f$ is of bounded variation iff the set of discontinuities of $f$ is finite. I didn't have problems with the forward ...
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1answer
244 views

Relation of total variation of a function $f$ and the integral of $|f'|$

Let $f:[a,b] \to \mathbb R$ a function of class $C^1$ on the interval $[a,b]$. Prove that: i) $f$ is a function of bounded variation. ii) The equality $V_a^b f= \int_a^b |f'(x)|dx$ holds. My ...
3
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1answer
167 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 ...
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2answers
64 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. ...
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2answers
507 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
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1answer
38 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., ...
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1answer
113 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 ...
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1answer
58 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 ...
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0answers
187 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 ...
8
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1answer
229 views

Question on Riemann sums

Question is : What I have read in Riemann sum definition is something like $$\sum_{i=1}^n f(y) .|x_i-x_{i-1}|$$ So, at first sight i am afraid this is not even related to Riemann integration of ...
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1answer
4k views

understanding upper bound and lower bound in lattice [closed]

i am studying discrete math. have a topic lattices, i really cant understand how to find greatest lower bound and lowest upper bound. any help would be appreciated.
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1answer
82 views

Is this functional differentiable?

A functional $\Phi$ is differentiable if there exist $F$ and $R$ such that $\Phi(f+h)-\Phi(f)=F(f,h)+R(f,h)$, where $F$ depends linearly on $h$ and $R(f,h) = O(h^2)$. Define a functional $\Phi(f) = ...
0
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1answer
99 views

Conditions for $o(|u|^{-1})$ decay of the Fourier transform of a bounded variation function

As the question suggests I am looking for a (not very restrictive) condition on a function of bounded variation so that its Fourier transform is $o(|u|^{-1})$ as $|u| \to \infty$. Let me elaborate on ...
0
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1answer
121 views

exponential upper bound on sum of exponentials

For what value of $c_3$ can I guarantee that $(a+b)\exp(-c_3\theta)>a\exp(-c_1\theta)+b\exp(-c_2\theta)$ where $a,b,c_i>0$
2
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1answer
163 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 ...
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0answers
92 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 ...
0
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1answer
370 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
32 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
370 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 ...
1
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1answer
63 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· ...
1
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4answers
152 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
votes
4answers
198 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 ...
1
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2answers
69 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
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1answer
241 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
63 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
48 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
155 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
88 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 ...
3
votes
1answer
244 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 ...
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1answer
50 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 $\mu=Du$ ?
1
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1answer
139 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
35 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
216 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 ...
0
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3answers
1k 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? $2\leq x <\infty$
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1answer
463 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
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1answer
120 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
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1answer
445 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
265 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
163 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
84 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
80 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
269 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
456 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
270 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 ...