Page 32 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 2: Functions of Bounded Variation

Notes
1 1 1
p
f (0) = 0 and f (x) = x cos px p 1 sin
x x 2 x
1 1
f (x) = x pxsin cos , for 0 < x 1
p–2
x x
f (x) is bounded for 0 x 1.
According to above problem, f BV [0, 1].

2.2 Summary

 A real-valued function f defined on [a,b] is said to be absolutely continuous on [a,b], if for
an arbitrary 0, however small, a, 0,s.t.

n n
f b r f a r whenever b r a r ,
r 1 r 1
where a b a b ... a b
1 1 2 2 n n
 A function f defined on an interval I is said to be monotonically non-increasing, iff

,
x y f x f y , x y . I

and monotonically non-decreasing, iff x > y f(x), f(y) x, g I.

n 1
b b b
 Let V f,P f x f x , and V f SupV f,P for all possible subdivisions P of
a r 1 r a a
r 0
b
[a,b]. If V f is finite, then f is called a function of bounded variation over [a,b].
a
2.3 Keywords

Absolute Continuous Function: A real valued function f defined on [a, b] is said to be absolutely
continuous on [a, b], if for an arbitrary > 0, however small, a, > 0, such that

n n
|f(b ) f(a )| , wherever (b a )
r r r r
r 1 r 1
where a < b a < b … a < b i.e. a ’s and b ’s are forming finite collection {(a , b ) : i = 1, 2,
1 1 2 2 n n 1 1 i i
…, n} of pair-wise disjoint intervals.
Continuous: A continuous function is a function f : X Y where the pre-image of every open
set in Y is open in X.

Disjoint: Two sets A and B are said to be disjoint if they have no common element, i.e. A B .

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