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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