Page 20 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 2: Functions of Bounded Variation

Notes
b
If V f is finite, then f is called a function of bounded variation or function of finite variation
a
over [a,b].
Set of all the functions of bounded variation on [a,b] is denoted by BV [a,b].

Notes

If f is defined on R, then we define

a
V f lin V f .
a a
Some important observations about the functions of bounded variations.

Let f: [a,b] R and P be any subdivision of [a,b]. Then:
x
(i) f x f a V f ,x [a,b]
a
a
(ii) V f 0
a
b b
(iii) P P V f,P V f,P , where P and P are any two subdivisions of [a,b].
1 2 1 2 1 2
a a
b b
(iv) V f,P V f , for all subdivisions P of [a,b].
a a
(v) For each 0, however small, at least one subdivision P’ of [a,b] such that
b b
V f,P' V f .
a a
b
(vi) V f 0.
a
b c
(vii) a b c V f V f .
a a
2.1.4 Theorems and Solved Examples

Theorem 1: A monotonic function on [a,b] is of bounded variation.
Proof: Divide the interval [a,b] by means of points

a x 0 x 1 x 2 ... x n b.
without any loss of generality, we can take f(x) as increasing function on [a,b]. Since if f is a
decreasing function, –f is an increasing function and so by taking –f = g, we see that g is an
increasing function and so we are allowed to consider only increasing functions. Thus

x x f x f x
r r 1 r r 1
f x r 1 – f x r 0
f x – f x f x – f x ...(i)
r 1 r r 1 r

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