Page 31 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes
Example: Show that the function

xsin if 0 x 2
f x x is continuous
0,if x 0

without being of bounded variation.
or

show that there exists a continuous function without being of bounded variation.

Solution: We know that lt f x 0 f 0
x 0

f x is continuous but not of bounded variation (see converse of above example.)
Hence the result.

Problem: Show that if f exists and is bounded on [a, b], then f BV [a, b].
Solution: According to given, let |f | M on [a, b].

Then for any X , x [a, b], we get
i – 1 i
f(x ) f(x ) M |f(x ) f(x )| M(x x )
i
i 1
x i x i 1 i i 1 i i 1
for any partition P of [a, b],

b
V(f) M (x i x ) M(b a)
i 1
a
f B V [a, b].

Problem: Show that the function f defined as

1
p
f(x) x sin for 0 x 1, f(o) 0, p 2.
x
is of bounded variation [0, 1].

1
p
(0 h) sin 0
Solution: Note that RF (0) = Lim h
h o h

1
= Limh (p 1) sin 0
h o h

1
p
( h) sin 0
h
and Lf (0) lim 0
h o h

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