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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2answers
55 views

Need guidance on a problem about oscillation from Spivak's Calculus on Manifolds

I've been stuck on this particular problem for a while now: Let $f: [a,b] \rightarrow \mathbb{R} $ be an increasing function. If $x_1, ... ,x_n \in [a,b]$ are distinct, show that $\sum\limits_{i=1}^n ...
2
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0answers
158 views

Arzela Ascoli variation theorem

I am trying to prove the following variation of the Arzel-Ascoli theorem: Let $(f_n)n$ be a sequence of differentiable functions on $[a,b]$ whose derivatives are uniformly bounded and there is an ...
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1answer
37 views

Definition of Bounded Variation Function with vectorial arguments

I try to found a definition of a function $$f(x)\colon\mathbb R^m\to \mathbb R^n$$ that use the norm. Is the formula below correct? $$TV=\sup\sum_{i=1}^k \|f(x_i)-f(x_{i-1})\|.$$ with k any finite ...
2
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1answer
61 views

Monotone convergence for monotone functions in BV

For $n \geq 1$ let $f, f_n : [0, 1] \to \mathbb{R}$ be monotone nonincreasing functions. Suppose that $f_n \nearrow f$ pointwise monotonically as $n \to \infty$. Is then $\mathrm{TV}(f - f_n) \to 0$ ...
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1answer
21 views

Trying to look for a lower bound on distance between distributions

I have two distributions $X$ and $Y$, is it true that for any arrangement of $X$ and $Y$ values $d= \sum_{i=1}^{n}(x_j-y_j)^2$ follows $d\geq\sum_{i=1}^{n}(x_i-y_i)^2$ where $x_i$ and $y_i$ are the ...
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2answers
41 views

Find limits of value/derivatives defining a polynomial at 2 points to bound it in between

The following properties define the polynomial $p(x)$ uniquely: $\text{deg}(p(x))=7\\p(-1)=y_1,\ p'(-1)=d_{1,1},\ p''(-1)=d_{2,1},\ p'''(-1)=d_{3,1}\\ p(1)=y_2,\ \ \ \ p'(1)=d_{1,2},\ \ \ \ ...
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1answer
33 views

Bounded & Norm space [closed]

Can someone help me on this exercise ? Thanks!
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1answer
311 views

Uniformly convergent subsequence of Equicontinuous R to R functions which are bounded in a point

Suppose that $(f_n)$ is an equicontinuous sequence of functions $f_n : \mathbb{R}\rightarrow \mathbb{R}$, such that $(f_n (0))$ is a bounded sequence in $\mathbb{R}$. Does there exist a subsequence ...
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0answers
23 views

A question on a function of bounded semivariation(part 2)

Let $(X,\tau)$ be a Hausdorff locally convex topological vector space and let $P(X)$ be a family of continuous seminorms on $X$ that generates the topology $\tau$. I got the following definition from ...
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1answer
28 views

Is a function of bounded semivariation bounded?

Let $X$ be a topological vector space and let $f:[a,b]\to X$. We say that $f$ is of bounded semi-variation in $[a,b]$ if the set $SV(f,[a,b])$ consisting of all the elements of the form $$\sum_{i=1}^n ...
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1answer
34 views

Partitions of $\alpha$ Variation

Suppose $T>0$ . Does anyone know if there exists a sequence of partitions $(\pi_n)_{n\in\mathbb{N}}$ of the interval $[0,T]$ such that the mesh size goes to $0$, and such that it is of bounded ...
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1answer
95 views

Total variation for functions, Meaning of supremum as used here?

On Wikipedia article, here: http://en.wikipedia.org/wiki/Total_variation, on definition 1.1 there says, "where the supremum runs over the set of all partitions ..." AFAIK supremum is defined for a ...
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3answers
4k views

If a function $f(x)$ is Riemann integrable on $[a,b]$, is $f(x)$ bounded on $[a,b]$?

Most statements regarding Riemann integrals (at least the ones that I have encountered) begin with the statement "for $f(x)$ bounded on $[a,b]$." I am wondering if Riemann integrability implies ...
0
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1answer
133 views

Finding the total variation of $3x^2-2x^3$ [closed]

I would appreciate if someone could help me to find the total variation of $3x^2-2x^3$ on $[-2,2]$. Thanks
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1answer
878 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 ...
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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
1k 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]?
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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
54 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
284 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
184 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
68 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
543 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 ...
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1answer
39 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
129 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
64 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
206 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 ...
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1answer
236 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
5k 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
84 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) = ...
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1answer
112 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
153 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
166 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
94 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 ...
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3answers
416 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 ...
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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 ...
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1answer
405 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
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· ...
2
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4answers
170 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
202 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
71 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
250 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
49 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 ...
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1answer
159 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 ...
4
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1answer
268 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
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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 $\mu=Du$ ?
2
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1answer
153 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 ...
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1answer
39 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 ...