Page 30 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 2: Functions of Bounded Variation

Notes
Example: A function f of bounded variation on [a,b] is necessarily bounded on [a,b] but
not conversely.

x b
Solution: If x [a,b], then f x f a V f V f
a a
b b
V f f x f a V f
a a
b b
f a V f f x V f f a
a a
f x is bounded on [a,b]

For the converse, define the function f on [0,1] by

0,if x 0
f x
x.sin ,if 0 x 1
x

since 0 x 1 and 1 sin 1, the function f is obviously bounded. Now consider the
x
partition

2 2 2 2 2
P= 0, , ,..., , , ,1 of [0,1]
2n+1 2n–1 7 5 3

Where n N. Then we get

1 2 2 2 2
V f,P f f 0 ... f f f 1 f
0 2n 1 3 5 3

2 n 2 2 2
1 0 ... 1 .1 0 1
2n 1 3 5 3

2 2 2 2
...
2n 1 3 5 3

1 1 1
4. ... .
3 5 2n 1

1
But we know that series is divergent. Therefore letting n we get that
2n 1

1 1
V f lt V f,P
0 n 0
f is not of bounded variation.

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