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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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 $?
13
votes
4answers
641 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 ...
9
votes
1answer
800 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 ...
9
votes
1answer
171 views

If a derivative $f^\prime$ is of bounded variation then prove that $f^\prime$ is continuous.

I am having trouble with the following problem: Let $f:[a,b]\to \mathbb R$ be differentiable on $[a,b]$ and $f^\prime$ is of bounded variation on $[a,b]$. Prove that $f^\prime$ is continuous ...
8
votes
1answer
244 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 ...
7
votes
3answers
6k 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 ...
7
votes
2answers
2k views

Does differentiable function of bounded variation have bounded derivative?

I learned that $f$ is a function of bounded variation, when function $f$ is differentiable on $[a,b]$ and has bounded derivative $f'$. What I want to know is converse part. If $f$ is differentiable ...
6
votes
1answer
910 views

Functions of bounded variation as the dual of $C([a,b])$

I am trying to understand this proposition about the dual of $C([a,b])$. I would like some help with the following: (1) What does the integral with respect to a function of bounded variation mean? ...
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 ...
6
votes
1answer
332 views

Absolute continuity

$f$ is continuous and of bounded variation on $[0,1]$, $f$ is absolutely continuous on any $[c,1]$ with $c \in (0,1]$. Then $f$ is absolutely continuous on $[0,1]$. How to show this? ...
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 ...
6
votes
1answer
183 views

Is this equivalent to bounded variation?

Let $\Omega \subset \mathbb{R}^n$ be open and bounded with smooth boundary. For $f \in L^1(\Omega)$, define $$ \|D_1 f\|_M(\Omega) =\inf\left\{\liminf_{k\to\infty}\int_\Omega |\nabla f_k|\,dx \mid ...
5
votes
1answer
84 views

questions regarding definition of bounded variation of several variables

$\int_\Omega u(x)\,\mathrm{div}\boldsymbol{\phi}(x)\mathrm{d}x = -\int_\Omega \langle\boldsymbol{\phi}, Du(x)\rangle \qquad \forall\boldsymbol{\phi}\in C_c^1(\Omega,\mathbb{R}^n)$ ...
4
votes
5answers
5k 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". ...
4
votes
2answers
648 views

a problem on functions of bounded variation

Which of the following statements are necessarily true ? a. Any continuous function on [$0, 1$] is of bounded variation. b. If $f : \mathbb{R} → \mathbb{R} $ is continuously differentiable, then its ...
4
votes
1answer
1k 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 ...
4
votes
1answer
3k 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 ...
4
votes
3answers
377 views

Total variation of (weakly) differentiable functions

the total variation of a function $u\in L^1(\Omega)$, $\Omega\subset \mathbb{R}^n$, can be defined as $$ \sup \{ \int_\Omega u \; \mathrm{div} g \; dx:\; g \in C_c^1(\Omega,\mathbb{R}^n), \; \lvert ...
4
votes
1answer
258 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 ...
4
votes
2answers
2k views

If a function $f$ is differentiable on $[a,b]$ and its derivative $f^\prime$ is integrable, must $f$ be of bounded variation?

I know there is a theorem saying if $f$ defined on $[a,b]$ is of bounded variation, then it is differentiable on $(a,b)$ a.e and $f'$ is integrable over $[a,b]$. I wonder whether the converse is ...
4
votes
1answer
33 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 ...
4
votes
2answers
185 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 ...
4
votes
1answer
526 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
1answer
1k views

Total Variation and Integral

Let $f$ be a function of bounded variation on $[a, b]$ and $T_{a}^{b}(f)$ its total variation. We do not assume that $f$ is continuous. Show that $$\int_{a}^{b}|f'(t)|\, dt \leq T_{a}^{b}(f).$$ ...
4
votes
1answer
255 views

Does the total variation of a function bound its numerical integration error, much like its first derivative?

When estimating the convergence of a Riemann sum to its integral, or equivalently the error in numerical integration, the commonly used bound is by upper bounding it's first derivative (see, for ...
4
votes
1answer
314 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 ...
4
votes
1answer
268 views

Weak derivatives

How can I prove that $|\nabla u|=|\nabla|u||$ when $u$ is regular enough for example Lipschitz or $W^{1,1}_{loc}$. Other question is about the pointwise derivative when $f:[0,1]\to R$ is BV is that ...
4
votes
1answer
124 views

Total variation of a Fourier series

Let $f(x) = f(x+2\pi)$ be a bounded real function given by the Fourier series of the form $$ f(x) = \sum_{k=1}^N a_k \sin(kx + \phi_k). $$ What is the total variation $V(f)$ of this function over one ...
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 ...
4
votes
1answer
376 views

Monotone increasing continuous function with $\int_a^b f' = f(b) - f(a)$ which is not absolutely continuous

If $f:[a, b] \to \mathbb{R}$ is continuous and real-valued, f' integrable on [a, b], and $\int_a^b f' = f(b) - f(a)$, must f be absolutely continuous? What if f is monotone increasing? For the ...
4
votes
1answer
124 views

Bounded variation problem

Let $x=0.a_1a_2\dots$ be the decimal expansion of a number $x$, $0<x<1$. If two decimal expansions of $x$ exist, the one that ends with $0$’s is taken. For what values of $q > 1$ is the ...
3
votes
4answers
213 views

Stability theory: Every solution of the scalar equation: $\ddot{x}+\left [a+b(t) \right ]x=0$ is bounded in $\left [t_0, +\infty \right )$.

EDITED Prove that: If $a>0$ and $$\int_{t_0}^{\infty} |b(t_1)|\mathrm{d}t_1<+\infty$$ then every solution of the scalar equation: $$\ddot{x}+\left [a+b(t) \right ]x=0$$ is bounded in $\left ...
3
votes
3answers
100 views

The limit of a sequence of uniform bounded variation functions in $L_1$ is almost sure a bounded variation function

Let $\{f_n\} $be a sequence of functions on $[a,b] $ that $\sup V^b_a (f_n) \le C$, if $f_n \rightarrow f $ in $L_1$ ,Prove that $f $ equals to a bounded variation function almost every where. I ...
3
votes
2answers
631 views

Exercise from Stein again - characterization of BV functions

I find this pretty hard and it would be awesome if someone could help me. The problem is the following (Problem 6/Chapter 3 from S&S's Real Analysis). Suppose $F$ is a bounded measurable ...
3
votes
2answers
156 views

If $f$ is of bounded variation on $[a+\epsilon, b]$, does it imply $f$ is of bounded variation on $[a,b]$?

The problem goes like: Suppose that $f\in B[a,b]$. If $V^b_{a+\epsilon}f\leq M$ for all $\epsilon >0$, does it follow that f is of bounded variation on $[a,b]$? I think the answer is yes. Since ...
3
votes
1answer
60 views

Show $BV[a, b]$ is not dense in $B[a, b]$

Show that $BV[a, b]$ is not dense in $B[a, b]$ under the metric $||f||_\infty$. I was wondering if I could get a hint.
3
votes
1answer
80 views

$f:[a,b]\to \mathbb R$ is continuous , has a finite number of local maxima and minima ; then how to prove that $f$ is bounded variation on $[a,b]$ ?

If $f:[a,b]\to \mathbb R$ is a continuous function having finite number of local maxima and minima ; then how to prove that $f$ is bounded variation on $[a,b]$ ?
3
votes
2answers
1k views

If f is of bounded variation is f Riemann integrable?

I want to know if f is of bounded variation on [a,b] does it follow that f is Riemann integrable on [a,b]?
3
votes
4answers
223 views

$\{z\in C:|z| = |\operatorname{re}(z)| +|\operatorname{im}(z)|\}$ open or closed [duplicate]

Possible Duplicate: Proving that a complex set in open/closed/neither and bounded/not bounded I think $\{z\in C:|z| = |\operatorname{re}(z)| +|\operatorname{im}(z)|\}$ is closed. But I have ...
3
votes
1answer
61 views

Show that $\,f(x) = \sum_{n=0}^\infty a_nx^n$, for $x \in [0,1]$, is of bounded variation

Let $\{a_n\} \subset \mathbb{R}$, be such that $\sum_{n=0}^\infty \lvert a_n\rvert < \infty$. Define $$f(x) = \sum_{n=0}^\infty a_nx^n \quad \text{for } x \in [0,1]$$ Prove that $f$ is of bounded ...
3
votes
1answer
546 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
211 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
273 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) ...
3
votes
1answer
170 views

Function coincides with a function of bounded variation almost everywhere

Problem Suppose $f\in L^1(\mathbb R)$ satisfies that there exists $A\ge0$ such that $$\int_{\mathbb R}\lvert f(x+h)-f(x)\rvert dx\le A\lvert h\rvert$$ for all $h\in\mathbb R$. We need to show that ...
3
votes
1answer
235 views

Continuity of bounded variation functions

A function $f:\mathbb{R}\rightarrow\mathbb{R}$ is a BV function if there exists $M<\infty$ for which $$\sum_{k=1}^N|f(x_k)-f(x_{k-1})|\leq M$$ for every sequence $x_0<x_1<\ldots<x_N$ and ...
3
votes
1answer
287 views

Functions of bounded variation and $L^\infty$ functions

Here are my questions. Are $L^\infty$ functions of bounded variation? Is the composition of two BV functions still of bounded variation? Is $x\mapsto \frac 1{f(x)}$ of bounded variation when $f$ ...
3
votes
1answer
72 views

Differentiable function in n dimensions

If a function $f:\mathbb R^{n} \rightarrow \mathbb R^m $ is differentiable at a point $a$ can we say that there is a neighbourhood of $a$ such that $f$ is locally Lipschitz? (i.e. there is some ...
3
votes
1answer
322 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.
3
votes
1answer
189 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 ...
3
votes
0answers
40 views

$\| f \|_{BV} = | f(a) | + V_{a}^{b} f$ defines a norm in the space $BV[a,b]$

I'm learning about functions of bounded variations and need to verify my work this problem: Show that $\| f \|_{BV} = | f(a) | + V_{a}^{b} f$ defines a norm in the space $BV[a,b]$. My attempt ...