The Mathematics of Financial Modelingand Investment Management

4-PrincipCalculus Page 105 Friday, March 12, 2004 12:39 PM

Principles of Calculus 105

EXHIBIT 4.4 Graphical Illustration of Right Continuous and Left Continuous

tion is not continuous at any point of [0,1] as its limit does not exist at

any point of its domain.

TOTAL VARIATION

Consider a function f(x) defined over a closed interval [a,b]. Then con-

sider a partition of the interval [a,b] into n disjoint subintervals defined

by n + 1 points: a = x 0 < x 1 < ... < xn–1 < xn = b and form the sum

n

T = ∑ f x()i – fx( i– 1 )

i = 1

The supremum of the sum T over all possible partitions is called the

total variation of the function f on the interval [a,b]. If the total varia-

tion is finite, the function f is said to have bounded variation or finite

variation. Note that a function can be of infinite variation even if the