“real: chapter_07” — 2011/5/22 — 23:09 — page4—#4
7-4 Real Analysis
7.3 PROPERTIES OF TOTAL VARIATION
Theorem 7.3.1
(i) If f is monotonic on [a, b], then f ∈ BV [a, b].
(ii) If f satisfies a Lipschitz condition of order 1, then f ∈ BV [a, b].
(iii) If f is continuous on [a, b] and f
exists and is bounded on the
interval (a, b), then f ∈ BV [a, b].
(iv) If f ∈ BV [a, b], then f is bounded.
Proof (i) For any partition P of [a, b] we have V (f , P) = f (b) −f (a)
or f (a) − f (b) depending on whether f is increasing or decreasing.
Hence the result.
(ii) Let |f (x) − f (y)|≤k|x − y| for x, y ∈[a, b]. For any partition
P ={t
0
, t
1
, t
2
, ..., t
n
} of [a, b], we have
V (f , P) =
n
i=1
|f (t
i
) − f (t
i−1
)|≤k
n
i=1
|t
i
− t
i−1
|
= k
n
i=1
(t
i
− t
i−1
) = k(b − a).
Thus V
b
a
f = sup ...