Transcription of Functions of Bounded Variation
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Functions of Bounded Variation
Functions of Bounded VariationOur main theorem concerning the existence of Riemann Stietjes integrals assures usthat the integral baf(x)d (x) exists whenfis continuous and is monotonic. Our lin-earity theorem then guarantees that the integral baf(x)d (x) exists whenfis continuousand is the difference of two monotonic Functions . In these notes,we prove that is thedifference of two monotonic Functions if and only if it is of Bounded Variation , whereDefinition 1(a) The function : [a, b] IR is said to be of Bounded Variation on [a, b] if and only ifthere is a constantM >0 such thatn i=1 (xi) (xi 1) Mfor all partitions IP ={x0, x1, , xn}of [a, b].(b) If : [a, b] IR is of Bounded Variation on [a, b], then the total Variation of on [a, b]is defined to beV (a, b) = sup{n i=1 (xi) (xi 1) IP ={x0, x1, , xn}is a partition of [a, b]}Example 2If : [a, b] IR is monotonically increasing, then, for any partition IP ={x0, x1, , xn}of [a, b]n i=1 (xi) (xi 1) =n i=1{ (xi) (xi 1)}= (xn) (x0) = (b) (a)Thus is of Bounded Variation andVf(a, b) = (b) (a).
Functions of Bounded Variation Our main theorem concerning the existence of Riemann–Stietjes integrals assures us that the integral Rb a f(x) dα(x) exists when f is continuous and α is monotonic.
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