 My Math Forum Total variation of a continuous function (of bv) is continuous
 User Name Remember Me? Password

 Real Analysis Real Analysis Math Forum

 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: $| T_f (x) - T_f (y) | = \sup \{ \sum |f(t_i) - f(t_{i-1})| \}$ where $y=t_0 \leq t_1 ... \leq t_n = x$. Then write f as the difference of 2 increasing functions so $\sup \{ \sum |f(t_i) - f(t_{i-1})| \} = \sup \{ \sum |(F(t_i) - G(t_i)) - (F(t_{i-1}) - G(t_{i-1})) | \} \leq \sup \{ \sum |F(t_i) - F(t_{i-1})| + |G(t_i) - G(t_{i-1}) | \} = |F(x) - F(y)| + |G(x) - G(y)|$ 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 Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post jiggs Real Analysis 1 December 5th, 2016 05:39 PM noobinmath Calculus 6 September 4th, 2015 10:20 PM natt010 Real Analysis 3 May 4th, 2014 02:51 PM thedoctor818 Real Analysis 17 November 9th, 2010 08:19 AM babyRudin Real Analysis 6 October 24th, 2008 12:58 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top

Copyright © 2019 My Math Forum. All rights reserved.       