# Tagged Questions

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### Total variation as surface area smooth functions of two variables.

I learnt we have different definitions for the total variation for functions of the form $f:\mathbb{R}^2\to\mathbb{R}$ which are in some way analogous to the total variation of functions of one ...
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### Books with exercises or problems on the space of functions of bounded variation

I am studying BV space (the space of functions of bounded variation) by using Evans & Gariepy's book. However, there are no exercises for BV space, and I have no idea where I can find some. ...
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### Why is the weakest bound only possible at these specified mins

Given the following: $$0 < g_0 \leq g_1$$ $$0 \leq B_0 \leq B_1$$ $$D_0 \leq D_1$$ (note: these are the only two variables that could be negative) $$0 \leq c_0$$ $$0 \leq c_1$$ $$0 < f_0$$ ...
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### Property of the variation of a function

I need help with the following: given $f:[a,b]\rightarrow \mathbb{R}$, show that $$V_f (a;b)=V_f(a;c)+V_f (c;b)$$ with $a< c <b$. We know that $$V_f(a;b) \geq \sum_i |f(x_i)-f(x_{i-1})|$$ for ...
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### Change of variables formula with integrator of bounded variation

Let $G$ be be continuous with bounded variation on finite intervals. If $f$ is continuous then it is well known that $\int_a^bf(G(s))dG(s)=\int_{G(a)}^{G(b)}f(x)dx$. How general can $f$ be so that ...
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### Unbounded variation but differentiable everywhere

A function with bounded variation is differentiable almost everywhere. There are also functions with unbounded variation that are differentiable almost everywhere (e.g. take ...
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### Question on Bounded Variation involving partitions

If $f\colon[a,b]\rightarrow \mathbb{R}$ and $P=(a=x_0, x_1, ...,x_n=b)$ is any partition of $[a,b]$ let $v_f(P)$ be defined as $$v_f(P) = \sum_{k=1}^n |f(x_k) - f(x_{k-1})|$$ and the total variation ...
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### Question on Functions of Bounded Variation

We say that $f$ is of bounded variation over $[a,b]$ or $f\in BV[a,b]$ if $V_f[a,b] = \sup \sum_{k=1}^n |f(x_k) - f(x_{k-1})| < \infty,$ where the supremum is taken over all possible partitions ...
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### Does bounded and continuous implies Lipschitz?

If a function $f : \mathbb{R} \rightarrow \mathbb{R}$ is integrable, bounded and continuous, is it also Lipschitz continuous?
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### Does the fundamental theorem of calculus hold for BV functions?

I am a bit confused and I hope you can help me in understanding a bit better these things. Let us start by considering one dimensional case. Let $f\colon \mathbb (a,b) \to \mathbb R$ be a function. ...
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### Understanding the duals of $L^{\infty}(K)$ and $C(K)$ and weak-* compactness

Let $(K\subset \mathbb{R}^n,\mathcal{B}(K),\lambda)$ be a measure space, where $\lambda$ is the Lebesgue measure and $K$ is compact. According to Wikipedia (with adapted notation), The dual ...
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### Continuity and Inverse Preservation of Boundedness Implies Preservation of Closed Sets

Sorry about the rather long title - wasn't sure what else to call it! Here is my question: Let $f : \Bbb R^n \rightarrow \Bbb R^m$ be continuous and such that $f^{-1}(F)$ is bounded whenever $F$ ...
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### Does bounded variation imply boundedness

Using the standard definition $$||f||_{TV} := \sup_{x_0<\cdots<x_n}\sum_{i=1}^{n} |f(x_i) - f(x_{i-1})|.$$ 1.When the domain is a bounded interval $[a,b]$, the statement holds. 2.When the ...
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### Function coincides with a function of bounded variation almost everywhere

Problem Suppose $f\in L^1(\mathbb R)$ satisfies that there exists $A\ge0$ such that $$\int_{\mathbb R}\lvert f(x+h)-f(x)\rvert dx\le A\lvert h\rvert$$ for all $h\in\mathbb R$. We need to show that ...
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### A question related to bounded variation

Let $f\in C^{BV}([0,1])$ (i.e. continuous and has bounded variation). Let the intervals $I$ and $T$ satisfy the following: $I\subset T\subset [0,1]$ and for sufficient small $\delta>0$, ...
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### 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 ...
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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 $$T^b_a(F) = \int^b_a ... 1answer 108 views ### About Regulated and Bounded Variations in Banach Spaces In the following definitions, we assumed that (X,\left\|\cdot\right\|) is a Banach space. Definition 1. f:[a,b]\to X is of bounded variation on [a,b] if ... 1answer 62 views ### if f_n\to f uniformly in [a,b] then f\in BV While preparing to test in calculus I found the question above: Let f_n(x)\to f uniformlly on [a,b]. Prove or give counterexample: if \forall n\in\mathbb N f_n(x)\in BV and \exists M>0 ... 1answer 275 views ### Two questions on Lebesgue Decomposition of an increasing function? I come up with the question in doing Stein's Real analysis, Chap3. Ex. 24, which assert that any increasing function f on [a,b] can be decomposed as$$F=F_A+F_C+F_J,$$with F_A is absolutely ... 1answer 336 views ### Bounded variation and \int_a^b |F'(x)|dx=T_F([a,b]) implies absolutely continuous If F is of bounded variation defined on [a,b], and F satisfies$$\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? ...
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 ...
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 ...