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?