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Topic: Question on bounded variation functions
Replies: 8   Last Post: Nov 9, 2009 9:07 AM

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miki

Posts: 47
Registered: 11/20/07
Question on bounded variation functions
Posted: Nov 7, 2009 4:18 AM
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Hello All,

The well-known definition of bounded variation functions is about
their behavior on closed intervals.
To say, "The total variation of real-valued function f, defined on an
interval [a, b] belongs to R is the quantity V_a_b_(f) = sup(sum(|f(x_i
+1) - f(x_i)|)) of all partitions of the interval considered", (etc.)

My question is, can I use or define total variation for a half-open
interval, say, [a, b)?

In any case, my intention is to define (or use) the following: for any
eps > 0 the total variation of the function
f on the interval [a, b - eps] is finite. Is it the same as to say
that the total variation of a function over the half-open interval [a,
b) is finite?


Regards,
Miki

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