For questions about functions $f$ defined on an interval $[a,b]$ such that there exists a constant $M>0$, such that if $a=x_0
2
votes
2answers
33 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.
...
0
votes
0answers
24 views
Integration of a $BV$ function with respect to a finite, signed Radon measure
Let $u,w\in BV(0,1)$ be given. Since we are in dimension 1 $u$ is continuous almost everywhere and has a representation $u^{l}$ and $u^{r}$ (left, right hand side continuous). Thus we can consider the ...
0
votes
0answers
15 views
Lemma 3.3 from “Positive solutions for third order semipositone boundary value problems”
I have this lemma :
Assume that : $w(t)$ is nondecreasing and $w(t)>0$ on $(q,1]$ , and let $M(t)$ such that $M\in L(0,1)$; $M(t)>0 $ on $(0,1)$ and $f(t,x+\gamma(t))\geq -M(t)$ for $(t,x)\in ...
1
vote
1answer
68 views
Taking the derivative of an integral of a discontinuous function
When I took measure theory with Frank Jones' books years ago, I did every problem in the book because I loved its teaching style.
There was one problem that took me 4-5 years to solve. It was problem ...
0
votes
0answers
18 views
Lemma 3.2 from “Positive solutions for third order semipositone boundary value problems”
How de prove this lemma please :
Assume that: $w(t)$ is nondercreasing and $w(t)>0$ on $(q,1]$ , $\frac12<p<q<1$ hods .
Let $z\in C^2[0,1]\cap C^3(0,1)$ satisfy $z'''(t)\geq 0$ 0n ...
3
votes
1answer
53 views
Bounded variation implies Borel measurable
Suppose that $f\colon[a, b] \to \mathbb{R}$ is a function of bounded variation. Show that $f$ is Borel measurable.
I was wondering if I could get a hint.
4
votes
2answers
49 views
Find f ae-differentiable with $f´\in L^1(0,1)$ but not in $BV$…
Here is a natural question which I didn't find in Measure Theory books:
Construct a continuous function $f(x)$ in $[0,1]$ with derivative at ae $x\in(0,1)$, and so that $f'(x)\in L^1(0,1)$, but such ...
1
vote
1answer
34 views
Is it true that $C_0^\ast[0,+\infty) = NBV_{loc}[0,+\infty)$
Any function from $BV_{\operatorname{loc}}[0,+\infty)$ defines a continuous linear functional on $C_0[0,+\infty)$. But is it true that any continuous linear functional on $C_0[0,+\infty)$ is given by ...
0
votes
1answer
34 views
Using Chernoff bound to analysis the Lazyselect algorithm
It's my homework of the course of randomized algorithm. In the textbook (Randomized Altorithm by Rajeev Motwani et.al.), the author analyzed this algorithm using Chebyshev bound, but are there any ...
0
votes
0answers
29 views
which condition says that $f$ is necessarily bounded variation
Which of the following condition below imply that the $f:[0,1]\to\mathbb{R}$ is necessarily Bounded Variation?
monotone;
continuous and monotone;
has derivative on $(0,1)$;
bounded derivative on ...
1
vote
1answer
86 views
Uniform limit of continuous functions bounded variation
Prove or disprove that if $f:[a,b]\rightarrow\mathbb{R}$ is the uniform limit of a sequence of continuous functions each of which is of bounded variation, then $f$ is of bounded variation on $[a,b].$
0
votes
1answer
47 views
Can I make a BV function right-continuous this way?
Math people:
This question is related to how can you "fix" one of the definitions of a BV function of one variable? . Suppose $f \in BV([0,1])$. I really have two-three questions. The ...
0
votes
0answers
26 views
Composition of bounded variation functions
Assume $f\in BV[a,b]$ and $g : [c,d]\rightarrow[a,b]$ is increasing, continuous, and onto. Prove that $F:=f\circ g\in BV[c,d]$ and $V^b_a f=V^d_c F$
2
votes
1answer
51 views
Proving Bounded variation is smallest linear space
Prove that $BV[a,b]$ is the smallest linear space containing all monotone functions on $[a,b].$
1
vote
0answers
18 views
Positive solutions for third order semipositone boundary value problems
please what is "semipositone"
http://www.sciencedirect.com/science/article/pii/S0893965909000500
thak you
