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 of $[a,b]$ My question is: If $f \in BV[a,b],$ show that $|f(x)|\leq |f(a)| + V_f[a,b] \ \ $ for all $x\in [a,b],$ […]

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 of the compact interval $[a,b]$. For function $g:[a,b]\to X$ and each $D\in \mathcal{D}$, we write $$V(g,D)=(D)\sum [g(v)-g(u)]$$ where $$D=\{[u,v]\}$$ […]

Related: When is it that $\int f d(g+h) \neq \int f dg + \int f dh$? In this context, I write “integration” to mean the Riemann-Stieltjes integeation Let $g:[a,b]\rightarrow \mathbb{R}$ be of bounded variation. Let’s define $\alpha(x)=V_a^x(g)$ and $g_1(x)= V_a^x (g) – g(x)$ and $g_2(x)= V_a^x(g) + g(x)$. ($V_a^x$ means the total variation) Then, these […]

Given any interval ${[a,b]}$, define the total variation ${\|F\|_{TV([a,b])}}$ of ${F}$ on ${[a,b]}$ as $$ \displaystyle \|F\|_{TV([a,b])} := \sup_{a \leq x_0 < \ldots < x_n \leq b} \sum_{i=1}^n |F(x_i) – F(x_{i+1})|. $$ Let $F:[a,b]\to\mathbb{R}$ be a continuous function. Can one conclude that $$\|F\|_{TV([a,b])}=\lim_{\epsilon\to 0+}\|F\|_{TV([a+\epsilon,b])}?$$ If $F$ is absolutely continuous than this can by done by […]

Prove that if $f$ is continuous on $[a,b]$ and $g$ is bounded variation on $[a,b],$ then $$\vert\int_a^bfdg\vert\le [sup_{a\le t \le b} \vert f(t) \vert] V_{[a,b]}g$$ Proof: As f is continuous on [a,b] and g is BV([a,b]) then f is riemann-stieltjes integrable, i.e. $f\in R(g)$. But I don’t know how to prove $\vert\int_a^bfdg\vert\le [sup_{a\le t \le […]

Consider $BV[a,b]$ the space of all bounded variation functions on a real interval $[a,b]$, endowed with the total variation norm $TV$. $AC[a,b]$, the space of absolutely continuous functions, is a subspace of $BV$. Is it closed?

What is the total variation of a dirac delta function $\delta(x)$? My guess is that it is something like $\infty$. If not defined, what would be the best way to define?

In Stephen Boyd’s book, Boyd uses the theorem that a linear function is bounded below on $R^m$ only when it is zero. I can’t really digest this. Can someone tell me why this holds? I mean if I take a line, convert it into a ray. It can start at any point. So indeed its […]

I’m trying to show that $f(x)$ is of bounded variation where $f(x)=x^{3/2}\sin(\frac{1}{x})$ on $[0,1]$. I think that it is but I can’t show it explicitly. Any help will be appreciated.

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 domain is $\mathbb{R}$ and the function is monotone, the statement holds both ways (if and only if). But what about in general? My guess is true, and here is my arguement: If […]

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