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May 13th, 2017, 11:07 PM   #1
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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.
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