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Questions tagged [bounded-variation]

For questions about functions $f$ defined on an interval $[a,b]$ such that there exists a constant $M>0$, such that if $a=x_0<x_1<\ldots<x_n=b$, $n\in\mathbb N^*$, then we have $\sum_{k=1}^n|f(x_k)-f(x_{k-1})|\leq M$.

This concept can be generalized to infinite intervals, requiring that the constant is uniform.

Let $[a,b]$ be a closed interval. A function $f\colon [a,b]\to \mathbb R$ is said to be of bounded variation if $$\sup_{n\in\mathbb N_{>0}}\sup_{a=x_0\lt x_1\lt\ldots\lt x_n=b}\sum_{j=1}^n|f(x_j)-f(x_{j-1})|<\infty.$$ We denote by $TV(f):=\sup_{n\in\mathbb N_{>0}}\sup_{a=x_0\lt x_1\lt\ldots\lt x_n=b}\sum_{j=1}^n|f(x_j)-f(x_{j-1})|$ the total variation of $f$, and we can endow the vector space of functions of bounded variation with the norm $\lVert f\rVert_{BV}:=TV(f)+|f(a)|$.

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Is $f(x) = \sqrt x$ of bounded variation?

I am trying to determine whether the $f(x) = \sqrt x$ is of bounded variation but only what I can proceed to is the definition of bounded variation: $$V_a^bf=\sup\left\{\sum_{i=1}^{n}\left|\sqrt{x_i} - \sqrt {x_{i-1}}\right|, n \in \Bbb N,…
delog
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Is Of Bounded Variation, only for the finite partition?

*prior to the body, note that title might be insufficient or inappropriate. Please edit it if it's needed. I am proving the claim below: Let $f: [a,b] \to \Bbb R$ be of bounded variation. $f(x) \ge c \gt 0$ for all $x \in [a, b]$ where $c$ is a…
Daschin
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$f$ is Of Bounded Variation $\Rightarrow$ $f$ is Bounded

Claim $f$ is Of Bounded Variation $\Rightarrow$ $f$ is Bounded Proof To prove above claim, I had seen the proof like below: "That $\|f\|_{TV}<\infty$ implies that $f$ is bounded is quite straightforward: $|f(x)|\le|f(0)|+|f(x)-f(0)|\le…
Daschin
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Hölder condition with Not of Bounded Variation

Let $\alpha$ $\in ]0, 1]$. $f : [a, b]$ $\to \Bbb R$ is said to satisfy a uniform Holder condition with exponent $\alpha$if there is some positive constant $M$ such that $\mid f(x_1) - f(x_2)\mid \lt M\mid x_1 - x_2\mid$, for all $x_1, x_2 \in [a,…
delog
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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?!
Mohammad W. Alomari
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image of a subset under a linear selfmap dense in the whole image

Given is a normed (and not necessarily complete) vector space $(X, ||\cdot||)$, a linear map $L:X\to X$ and a subset $D\subset X$. What conditions must $D$ satisfy such that $L(D)$ is dense in $L(X)$? This problem has the following background: Let…
sranthrop
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Prove that if $f$ is of bounded variation on $[a, b]$, then $f$ is bounded on $[a, b]$.

Prove that if $f$ is of bounded variation on $[a, b]$, then $f$ is bounded on $[a, b]$. My professor proved the proposition like the following processes: Choose $x$ between $a$ and $b$, that is, $x\in(a, b)$. let $\Gamma$ be a partition having…
Danny_Kim
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First Order Variation Example

I am in a introductory stochastic calculus class and came across and example from that asks for the first order variation and the quadratic variation of a continuous function. For example: $f(x) = cos(2x)$ defined on the interval $-\pi/2 <= x <=…
klib
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The convergence in Bounded Variation functions

Given a sequence $u_n\in BV(\Omega)$ and $u\in BV(\Omega)$, where $\Omega\in R^n$ is open. We assume $u_n\to u$in $L^1_{loc}(\Omega)$ and we also assume that $$ \lim_{n\to\infty}|D u_n|(\Omega)=|Du|(\Omega).$$ Now, give another open set $V\subset…
spatially
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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 $f_1$ and $g_1$ such that $f \leq f_1 \leq g \leq…
Keneth Adrian
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How do I find the percentage change of partial variation?

Z partly varies as x^2 and partly varies directly as y.If x is decreased by 20% and y is decreased by 36%, find the percentage change. I tried to do this with how I would do percentage changes in joint variation, but I just couldn't put the numbers…
user1039203
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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) as the same, although (1) has the drawback that you can…
Stefan Smith
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Limit and derivative

The function $f:\Bbb R\to\Bbb R$ is differentiable and $\lim_{x\to\infty}f'(x)=1$. Which of the following is true? A) $f$ is bounded B) $f$ is increasing C) $f$ is unbounded D) $f'(x)$ is bounded
user462999
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$f, g : [a, b] \to R$ be of bounded variation. Then $fg$ is of bounded variation

Claim $f, g : [a, b] → R$ be of bounded variation. Then show that $fg$ is of bounded variation. To prove above claim, I would like to derive the fact such as $\mid f(x_i) g(x_i)-f(x_{i-1})g(x_{i-1})\mid\le\mid f(x_i) -f(x_{i-1})\mid \mid…
Daschin
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A Version of Weiestrass Function : Which is easy to prove Satisfying Holder Condition and Not Of BV

I am looking for the function that satisfies two characteristics - first, satisfying the holder condition and the second, not of BV. One recommended me a Weiestrass functiuon, however, most generalized version of Weiestrass is too complex to prove…
Daschin
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