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 […]

Let $I=[a,b]$ with $a<b$ and let $u:I\rightarrow\mathbb{R}$ be a function with bounded pointwise variation, i.e. $$Var_I u=\sup\{\sum_{i=1}^n|u(x_i)-u(x_{i-1})|\}<\infty$$ where the supremum is taken over all partition $P=\{a=x_0<x_1<…<x_{n-1}<b=x_n\}$. How can I prove that if $u$ satisfies the intermediate value theorem (IVT), then $u$ is continuous? My try: $u$ can be written as a difference of two increasing […]

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 $$ T^b_a(F) = \int^b_a |f(t)|dt. $$ Now, here is what I have so far. Show $F$ is of BV is simple: let $a=x_0 < x_1 < \dots […]

Let $a, b \in \mathbb{R}$ with $a < b$ and define the compact interval $I := [a, b]$. Let $g, h : \mathbb{R} \rightarrow \mathbb{R}$ be non-decreasing and right-continuous on $I$ and constant on $(-\infty, a]$ and on $[b, \infty)$, separately, and define $f := g – h$, so that, over every compact interval, $g […]

What is the connection between finite Borel sign-changing measures and the functions with bounded variation on the same interval? Proof would be appreciated.

I am trying to find out the relation between those spaces. Take $I\subset R$ on the real line. $I$ can be unbounded. Then I have: We first assume $I$ is bounded. If $u\in C^{0,\alpha}(I)$, for $0<\alpha<1$, then $u$ is uniformly continuous for sure. However, I can not prove $u\in C^{0,\alpha}(I)$ then $u\in AC(I)$, nor $u\in […]

Let $f\colon [a,b] \to \mathbb R$ be of bounded variation. Must it be the case that $|\int_a ^b f’ (x) |\leq |TV(f)|$, where $TV(f)$ is the total variation of $f$ over $[a,b]$? If so, how can one prove this? In the standard proof of the monotone differentiation theorem, it is shown tat this holds for […]

I don’t seem to find version of this problem in the site, but I am sure this is pretty standard type of question. $f$ be of bounded variation on $[0,1]$, and $f$ is absolutely continuous (AC) on $[\varepsilon,1]$ for all $\varepsilon >0$ and $f$ is continuous at $0$. Now the goal is to prove $f$ […]

I’m looking for a continuous function $f$ defined on the compact interval $[0,1]$ which is not of bounded variation. I think such function might exist. Any idea? Of course the function $f$ such that $$ f(x) = \begin{cases} 1 & \text{if $x \in [0,1] \cap \mathbb{Q}$} \\\\ 0 & \text{if $x \notin [0,1] \cap \mathbb{Q}$} […]

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 that there exist a function $f$ that is bounded on set A? Can I prove it this way: Since $f_n […]

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