
Real Analysis Real Analysis Math Forum 
 LinkBack  Thread Tools  Display Modes 
May 13th, 2017, 11:07 PM  #1 
Member Joined: Oct 2013 Posts: 36 Thanks: 0  Total variation of a continuous function (of bv) is continuous
There don't seem to be many proofs of this in the books so I was wondering if this was correct: $ T_f (x)  T_f (y)  = \sup \{ \sum f(t_i)  f(t_{i1}) \}$ where $y=t_0 \leq t_1 ... \leq t_n = x$. Then write f as the difference of 2 increasing functions so $\sup \{ \sum f(t_i)  f(t_{i1}) \} = \sup \{ \sum (F(t_i)  G(t_i))  (F(t_{i1})  G(t_{i1}))  \} \leq \sup \{ \sum F(t_i)  F(t_{i1}) + G(t_i)  G(t_{i1})  \} = F(x)  F(y) + G(x)  G(y)$ since they're increasing. Then F and G are continuous so take $\delta$ to be $min \{ \delta_F , \delta_G \}$. Edit: nevermind I realised that F and G don't have to be continuous... Last edited by fromage; May 13th, 2017 at 11:43 PM. 

Tags 
continuous, function, total, variation 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
h(y) is a continuous function of y  jiggs  Real Analysis  1  December 5th, 2016 05:39 PM 
Is the function continuous?  noobinmath  Calculus  6  September 4th, 2015 10:20 PM 
Absolutely continuous/bounded variation  natt010  Real Analysis  3  May 4th, 2014 02:51 PM 
Proving that a space is continuous, continuous at 0, and bdd  thedoctor818  Real Analysis  17  November 9th, 2010 08:19 AM 
continuous at any point iff continuous at origin  babyRudin  Real Analysis  6  October 24th, 2008 12:58 AM 