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Real Analysis
book

Real Analysis

by V. Karunakaran
May 2024
Intermediate to advanced content levelIntermediate to advanced
585 pages
15h 38m
English
Pearson India
Content preview from Real Analysis
“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
i1
)|≤k
n
i=1
|t
i
t
i1
|
= k
n
i=1
(t
i
t
i1
) = k(b a).
Thus V
b
a
f = sup ...
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Publisher Resources

ISBN: 9781299447561Publisher Website