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May 14th, 2017, 12:07 AM  #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 14th, 2017 at 12:43 AM. 

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continuous, function, total, variation 
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