“real: chapter_07” — 2011/5/22 — 23:09 — page 20 — #20
7-20 Real Analysis
are increasing on [a, b]. By the previous theorem, we now know
that v(x) is absolutely continuous. Since the difference of two abso-
lutely continuous functions is again absolutely continuous, our result
follows.
7.7 GENERALIZATIONS
The concept of a function of bounded variation on a finite interval [a, b]
can be extended to complex-valued functions defined on the whole of
R.
Definition 7.7.1 Let f :
R → C be a function. We say that f is of
bounded variation on
R if
V (f ) = lim
x→∞
V
x
−∞
f
exists finitely where V
x
−∞
f = sup
P
n
i=1
|f (x
i
) − f (x
i−1
)|. Here P varies
over all choices of the set {x
i
/ i = 0, 1, 2, ..., n} with −∞ < x
0
<
x
1
< x
2
< ··· < x
n
= x. In this case, V (f ) will also be called ...