Fdtxjr

Is the destination of a commercial flight important for the pilot?

Is there any reason not to eat food that's been dropped on the surface of the moon?

Why "be dealt cards" rather than "be dealing cards"?

Will it be accepted, if there is no ''Main Character" stereotype?

The plural of 'stomach"

when is out of tune ok?

What is the term when two people sing in harmony, but they aren't singing the same notes?

Is it okay / does it make sense for another player to join a running game of Munchkin?

Is expanding the research of a group into machine learning as a PhD student risky?

What will be the benefits of Brexit?

Mapping a list into a phase plot

Efficiently merge handle parallel feature branches in SFDX

voltage of sounds of mp3files

Personal Teleportation as a Weapon

What would happen if the UK refused to take part in EU Parliamentary elections?

Greatest common substring

How to verify if g is a generator for p?

What is the opposite of 'gravitas'?

How will losing mobility of one hand affect my career as a programmer?

Can I use my Chinese passport to enter China after I acquired another citizenship?

How was Earth single-handedly capable of creating 3 of the 4 gods of chaos?

Tiptoe or tiphoof? Adjusting words to better fit fantasy races

Star/Wye electrical connection math symbol

Hostile work environment after whistle-blowing on coworker and our boss. What do I do?

Can we construct a function $g$ and a partition $pi$ such that $pi(g)

First order variation and total variation of a function/stochastic processClosure of space of simple functions in $L^infty([a,b])$Sub-additivity of the càdlàg continuity modulusLimit of arithmetic meansProve that $(k_n)$ and $(t_n)$ converge to the same limit.$int_a^b Vert gamma'(t)Vert dt=lim_delta(P)to0sum_k=0^nVert gamma(t_k+1)-gamma(t_k)Vert$The intersection of all the R-S integrable functions is the set of continuous functionsEvery polynomial $p(x)$ is of boundary variation in the interval $[a,b]$ and $V_a^b p leq int_a^b vert p'(s) ds vert$Show that the sequence is increasing and unboundedFor any continuous functions g in [0, 1], is there exist a partition sequence such that $[g, g]_pi$ = 0?

0

$begingroup$

Let g be a function defined on [a,b],
and partition $triangle= a=t_0<t_1<dots<t_n=b$
$delta(triangle)=displaystylemax_1 leq k leq n(t_k+1-t_k)$

and $pi$ be a partition sequence, that is
$pi=triangle_n:triangle_nsubseteqtriangle_n+1, delta(triangle_n)to0$.

Let the variation of $g$ over $triangle$ be $triangle(g)=sum_k=0^n-1(g(t_k-1)-g(t_k))$

and let $pi(g)=limlimits_nto inftytriangle_n(g)$
and let $v(g)=displaystylesup_triangletriangle(g)$.

(a) Can we construct a function g and partition $pi$ such that $pi(g)<v(g)<infty$?

(b) If $g$ is either right or left continuous at each point of [a,b], is $pi(g)=v(g)$ for any $pi$?

real-analysis

share|cite|improve this question

edited Mar 17 at 12:24

Bernard

123k741117

asked Mar 17 at 12:11

mathgogomathgogo

1

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I think an absolute value sign is missing in your definition of $Delta (g)$.
  $endgroup$
  – Kavi Rama Murthy
  Mar 17 at 12:54
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Any strictly increasing function meets the requirements in a). The answer to b) is NO.
  $endgroup$
  – Kavi Rama Murthy
  Mar 17 at 12:56
  
  

  
  
  
  

add a comment | 

0

$begingroup$

Let g be a function defined on [a,b],
and partition $triangle= a=t_0<t_1<dots<t_n=b$
$delta(triangle)=displaystylemax_1 leq k leq n(t_k+1-t_k)$

and $pi$ be a partition sequence, that is
$pi=triangle_n:triangle_nsubseteqtriangle_n+1, delta(triangle_n)to0$.

Let the variation of $g$ over $triangle$ be $triangle(g)=sum_k=0^n-1(g(t_k-1)-g(t_k))$

and let $pi(g)=limlimits_nto inftytriangle_n(g)$
and let $v(g)=displaystylesup_triangletriangle(g)$.

(a) Can we construct a function g and partition $pi$ such that $pi(g)<v(g)<infty$?

(b) If $g$ is either right or left continuous at each point of [a,b], is $pi(g)=v(g)$ for any $pi$?

real-analysis

share|cite|improve this question

edited Mar 17 at 12:24

Bernard

123k741117

asked Mar 17 at 12:11

mathgogomathgogo

1

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I think an absolute value sign is missing in your definition of $Delta (g)$.
  $endgroup$
  – Kavi Rama Murthy
  Mar 17 at 12:54
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Any strictly increasing function meets the requirements in a). The answer to b) is NO.
  $endgroup$
  – Kavi Rama Murthy
  Mar 17 at 12:56
  
  

  
  
  
  

add a comment | 

$begingroup$

Let g be a function defined on [a,b],
and partition $triangle= a=t_0<t_1<dots<t_n=b$
$delta(triangle)=displaystylemax_1 leq k leq n(t_k+1-t_k)$

and $pi$ be a partition sequence, that is
$pi=triangle_n:triangle_nsubseteqtriangle_n+1, delta(triangle_n)to0$.

Let the variation of $g$ over $triangle$ be $triangle(g)=sum_k=0^n-1(g(t_k-1)-g(t_k))$

and let $pi(g)=limlimits_nto inftytriangle_n(g)$
and let $v(g)=displaystylesup_triangletriangle(g)$.

(a) Can we construct a function g and partition $pi$ such that $pi(g)<v(g)<infty$?

(b) If $g$ is either right or left continuous at each point of [a,b], is $pi(g)=v(g)$ for any $pi$?

real-analysis

share|cite|improve this question

edited Mar 17 at 12:24

Bernard

123k741117

asked Mar 17 at 12:11

mathgogomathgogo

1

$endgroup$

share|cite|improve this question

edited Mar 17 at 12:24

Bernard

123k741117

asked Mar 17 at 12:11

mathgogomathgogo

1

share|cite|improve this question

edited Mar 17 at 12:24

Bernard

123k741117

asked Mar 17 at 12:11

mathgogomathgogo

1

edited Mar 17 at 12:24

Bernard

123k741117

edited Mar 17 at 12:24

Bernard

123k741117

asked Mar 17 at 12:11

mathgogomathgogo

1

asked Mar 17 at 12:11

mathgogomathgogo

1