# Tagged Questions

32 views

### 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 ...
30 views

### 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 ...
30 views

### Does bounded variation and continuous means total variation continuous

$F$ is of bounded variation and continuous. Is it true that total variation is continuous ? In case, $F$ is absolutely continuous it is trivial to see. But for the above case how to proceed ?
41 views

### One inequality involving Total variation function

If $F$ is of bounded variation in $[a,b]$, then I need to prove that $$\int_{a}^{b}|F'(x)| dx \leq T_F(a,b)$$ If $F'$ were Riemann integrable then it was easy to prove (in fact we can prove ...
133 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 ...
158 views

### Functions of bounded variation on all $\mathbb{R}$
Consider $F:\mathbb{R}\rightarrow\mathbb{R}$ such that $\sup_{a,b}T_F (a,b)<\infty$ where $T_F (a,b)$ is the total variation of $F$ on the interval $[a,b]$. Then we have i) ...
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 ...