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I have been doing some excercises on total variation when the following questions came up to my mind:

(1) Let $f$ be continuous on the interval $[0,1]$ and be of bounded variation. Is it true that its total vatiation function $TV(f_{[0,x]})$ is uniformly continuous? i.e. Is it true that for $\forall\space\epsilon>0$, $\exists\space\delta>0$, such that for arbitrary interval $[a,b]$ with $|b-a|<\delta$, we have $TV(f_{[a,b]})<\epsilon$?

(2) Alternatively, if it is not uniformly continuous, can I say that for $\forall\space{}x\in[0,1]\text{ and } \forall\space\epsilon>0$ , $\exists\space\text{ nondegenerate interval }I \text{ such that }x\in{}I\text{ and } TV(f_{I})<\epsilon$?

Thank you!

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Hint: What is the total variation function of a monotonous function? –  Did May 12 '12 at 11:46
    
@Didier I think it should be the absolute value of the difference of end point values? But I still don't see how I can use that on this question... –  Vokram May 12 '12 at 12:16
    
Hence, for example, if $f(0)=0$ and $f$ is nondecreasing and if $g(x)=$TV$(f_{[0,x]})$, then $g=\ldots$ –  Did May 12 '12 at 12:53
    
@Didier yeah in that case $g$ would be continuous. Hmm but sorry if I'm getting your point really slow here, I did not assume f is monotone in the question. Are you saying that I should use Jordan's decomposition and express it as the difference of two such functions? But I think that way I will get an inequality instead of an equality and there's no guarantee they are continuous any more –  Vokram May 12 '12 at 13:10
    
This shows is that the regularity of $x\mapsto$TV$(f_{[0,x]})$ is at most the regularity of $f$, in general. –  Did May 12 '12 at 13:31
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up vote 5 down vote accepted

Since a continuous function on a compact set is uniformly continous, you're actually asking whether the total variation of a continuous function is continuous.

Let $\epsilon\gt0$ and $x_0\in(0,1]$ be given. Since the total variation is non-decreasing, it suffices to find a point $x\lt x_0$ with $TV(f_{[x,x_0]})\lt\epsilon$ to show that the total variation is left-continuous, and thus by symmetry also right-continuous and hence continuous.

Pick some point $x_1\lt x_0$. By definition there is a partition of $[x_1,x_0]$ such that the sum of absolute differences of function values over the partition is within $\epsilon/2$ of $TV(f_{[x_1,x_0]})$. Since $f$ is continuous, we can find a point $x$ between the last intermediate point of the partition and $x_0$ such that $|f(x)-f(x_0)|\lt\epsilon/2$. Refining the partition with this point doesn't decrease its sum of absolute differences. Now the sum of absolute differences in the partition up to $x$ is within $\epsilon/2+\epsilon/2=\epsilon$ of $TV(f_{[x_1,x_0]})$, and thus so is $TV(f_{[x_1,x]})$; hence $TV(f_{[x,x_0]})\lt\epsilon$ as required.

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Nice! However, I had hard time understanding the last sentence, so maybe it is worth to supply a few more details as follows: Let $V$ denote the variation over the partition chosen within $\epsilon/2$ of $TV(f_{[x_1,x_0]})$ as described above. Then $$TV(f_{[x_1,x_0]})<V+\epsilon/2.$$ Also, $V-|f(x)-f(x_0)|$ is some variation over interval $[x_1,x]$, so we have: $$TV(f_{[x_1,x]})\ge V-|f(x)-f(x_0)|.$$ Finally, $TV(f_{[x_1,x_0]})=TV(f_{[x_1,x]})+TV(f_{[x,x_0]})$, so that: $$TV(f_{[x,x_0]}) = TV(f_{[x_1,x_0]}) - TV(f_{[x_1,x]})<(V+\epsilon/2) - (V-|f(x)-f(x_0)|)<\epsilon/2+\epsilon/2=\epsilon.$$ –  mathreader Jul 7 '13 at 0:09
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