0
votes
1answer
42 views

Problem on Bounded Variation

Assume $f$ is of bounded variation on $[a,b]$. Show that there is a sequence of partitions $\{P_n\}$ of $[a,b]$ for which the sequence $\{TV(f,P_n)\}$is increasing and converges to $TV(f)$? ...
1
vote
2answers
290 views

Is there an unbounded function with a bounded derivative?

I know that there exists bounded functions with unbounded derivatives. For example, $\sin(e^x)$ is bounded and differentiable everywhere on $\mathbb{R}$, but its derivative is unbounded. Is it ...
2
votes
2answers
75 views

Prove that this function has finite variation

We know that $f$ has finite variation on $[a,b]$. Prove that $$g(x)= \begin{cases} 0, & x=a\\[8pt] \frac{1}{x-a} \int _{a} ^x f(t) \, dt , & x \in (a,b] \end{cases} $$ has finite variation. ...
1
vote
2answers
283 views

Continuously differentiable functions of bounded variation

From this question, we know that a continuous function of bounded variation is not necessarily absolutely continuous. But the example (Devil's staircase) given is not differentiable. What if we ...
4
votes
1answer
307 views

An upper bound for an integral.

I want to find an upper bound of the function $h(t)$ on $t \in [0, \infty [$ with $d >1$ which is defined by $$ h(t) := \int_0^t (1+t)^d (1+t-r)^{-d} (1+r)^{-d} dr$$ and I could easily prove that ...
4
votes
0answers
921 views

Integral of the derivative of a function of bounded variation

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 ...