4-PrincipCalculus Page 106 Friday, March 12, 2004 12:39 PM
106 The Mathematics of Financial Modeling and Investment Management
function itself remains bounded. For example, the function that assumes
the value 1 on rational numbers and 0 elsewhere is of infinite variation
in any interval, though the function itself is finite.
Continuous functions might also exhibit infinite variation. The follow-
ing function is continuous but with infinite variation in the interval [0,1]:
0 for x =^0
fx()= π --- x
xsinfor 0 ≤<^1
x
DIFFERENTIATION
Given a function y = f(x) defined on the open interval (a,b), consider its
increments around a generic point x consequent to an increment h of the
variable x ∈(a,b)
∆y = f(x + h) – f(x)
Consider now the ratio ∆y/h between the increments of the depen-
dent variable y and the independent variable x. Called the difference
quotient, this quantity measures the average rate of change of y in some
interval around x. For instance, if y is the price of a security and t is
time, the difference quotient
yt ( + h)– yt()
∆y= ------------------------------------
h
represents the average price change per unit time over the interval
[t,t+h]. The ratio ∆y/h is a function of h. We can therefore consider its
limit when h tends to zero.
If the limit
fx ( + h)– fx()
f ′()x = lim ------------------------------------
h → (^0) h
exists, we say that the function f is differentiable at x and that its deriv-
ative is f ′,also written as
df dy
-------or -------
dx dx