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

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10
votes
3answers
8k views

If a function $f(x)$ is Riemann integrable on $[a,b]$, is $f(x)$ bounded on $[a,b]$?

Most statements regarding Riemann integrals (at least the ones that I have encountered) begin with the statement "for $f(x)$ bounded on $[a,b]$." I am wondering if Riemann integrability implies ...
4
votes
1answer
4k views

Bounded variation, difference of two increasing functions

Prove that if $f$ is of bounded variation in $[a,b]$, it is the difference of two positive, monotonic increasing functions; and the difference of two bounded monotonic increasing functions is a ...
13
votes
5answers
2k views

Question about Riemann integral and total variation

Let $g$ be Riemann integrable on $[a,b]$, $f(x)=\int_a^x g(t)dt $ for $x \in[a,b]$. Can I show that the total variation of $f$ is equal to $\int_a^b |g(x)| dx $?
5
votes
1answer
2k views

Is the total variation function uniform continuous or continuous?

I have been doing some excercises on total variation when the following questions came up to my mind: (1) Let $f$ be continuous on the interval $[0,1]$ and be of bounded variation. Is it true that ...
10
votes
1answer
989 views

Two definitions of “Bounded Variation Function”

As far as I know, a function $f$ defined on an interval $[a, b]$ is said to be of bounded variation if $$\tag{1}V_a^b(f)=\sup\left\{\sum_{P} \lvert f(x_{j+1})-f(x_j)\rvert \ :\ P\ \text{partition of }...
6
votes
5answers
6k views

Bounded functions have bounded derivatives.

Can the graph of a bounded function ever have an unbounded derivative? I want to know if $f$ has bounded variation then its derivative is bounded. The converse is obvious. I think the answer is "yes". ...
2
votes
1answer
303 views

The question regrading to density argument in analysis.

I know the density argument in $L^p$ space, in Sobolev spaces, and even in BV are really sweet in many many cases. However, some times author just work on nice functions and comment by "the rest can ...
0
votes
1answer
166 views

If a function is not continuous then is it possible for bounded function in given range? [duplicate]

I posted this problem before this .I have satisfied explaination given by markus-scheuer sir and siminore sir . I also found here . I have read the Wikipedia posts for continuous function and bounded ...
3
votes
1answer
276 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) $\int_\mathbb{R}|F(x+...
3
votes
1answer
605 views

Bounded variation and $\int_a^b |F'(x)|dx=T_F([a,b])$ implies absolutely continuous

If $F$ is of bounded variation defined on $[a,b]$, and $F$ satisfies $$\int_{a}^b |F'(x)|dx=T_F([a,b])$$ where $T_F([a,b])$ is the total variation. How to prove that $F$ is absolutely continuous? ...
3
votes
1answer
77 views

Does differentiability imply having bounded variation on some subinterval?

Suppose that $f:(a,b)\to\mathbb{R}$ is a differentiable function. Does it follow that $f$ has bounded variation on some subinterval $[c,d]\subset (a,b)$? Details and ideas Being differentiable ...
6
votes
1answer
2k views

Prove the normed space of bounded variation functions is complete

Let $\Vert f \Vert = |f(0)| + \mathrm{Var}f$ for all $f \in BV([0,1])$; we are given that it is a norm. Show that $BV([0,1])$ is a complete normed space with this norm. I have shown that any Cauchy ...
13
votes
4answers
1k views

A continuous function on $[0,1]$ not of bounded variation

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 ...
6
votes
1answer
1k views

proving $f$ is absolutely continuous on $[0,1]$

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 $[\...
8
votes
1answer
246 views

Question on Riemann sums

Question is : What I have read in Riemann sum definition is something like $$\sum_{i=1}^n f(y) .|x_i-x_{i-1}|$$ So, at first sight i am afraid this is not even related to Riemann integration of $f$...
4
votes
1answer
2k 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 ...
3
votes
1answer
264 views

Total variation measure vs. total variation function

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 $(-...
1
vote
0answers
28 views

The convergence of $p$-laplace equation

Let $g\in L^\infty(\Omega)$ be given, where $\Omega\subset \mathbb R^2$ is open bounded with smooth boundary. Define, for $1<p\leq 2$, $$ u_p:=\operatorname{argmin}\left\{\int_\Omega|u-g|^pdx+\...
5
votes
1answer
301 views

Unbounded variation but differentiable everywhere

A function with bounded variation is differentiable almost everywhere. There are also functions with unbounded variation that are differentiable almost everywhere (e.g. take $f:[0,1]\rightarrow\...
4
votes
1answer
370 views

Total Variation and indefinite integrals

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(...
4
votes
1answer
275 views

Bounded Variation $+$ Intermediate Value Theorem implies Continuous

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 ...
3
votes
1answer
209 views

Math Analysis - Problem dealing with bounded variation

Let $f\colon[0,1] \rightarrow \mathbb R$ (all real numbers) be defined by $\displaystyle f(x) = x \sin \left(\frac{\pi}{2x}\right)$ if $0 \lt x \le 1$ and $f(x) = 0$ if $x=0$. Determine whether ...
2
votes
1answer
197 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 $$\sup\left\{\sum|f(b_k)-f(a_k)|...
2
votes
1answer
503 views

Does bounded variation imply boundedness

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 ...
2
votes
1answer
273 views

About functions of bounded variation

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 ...
2
votes
1answer
48 views

Necessary and sufficient condition for Integrability along bounded variation

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 ...
1
vote
1answer
46 views

If $f,f',f''$ are bounded a.e., is $f'$ of bounded variation everywhere?

Assume the function $f$ is such that everywhere except in $0$: $f$ is bounded on $\mathbb{R}$ $f$ is twice differentiable everywhere except in $0$ $f'$ and $f''$ are bounded everywhere except in $0$ ...
1
vote
1answer
76 views

A question on a function of bounded semivariation

Let $(X,\tau)$ be a Hausdorff locally convex TVS and let $P(X)$ be a family of continuous seminorms on $X$ that generates the topology $\tau$. I got the following definition from one of the articles ...
0
votes
1answer
140 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
1answer
146 views

Is $x^{3/2}\sin(\frac{1}{x})$ of bounded variation?

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.
0
votes
1answer
123 views

how can you “fix” one of the definitions of a BV function of one variable?

Math people: My question is similar to that in Two definitions of "Bounded Variation Function" . If you look at that question, you will notice that people treat definitions (1) and (2) ...