Daniel J. Duffy - Numerical Methods in Computational Finance

An example of a function that is continuous and nowhere differentiable is the Weierstrass function that we can write as a Fourier series :

(1.6)

b is a positive odd integer and .

This is a jagged function that appears in models of Brownian motion. Each partial sum is continuous, and hence by the uniform limit theorem (which states that the uniform limit of any sequence of continuous functions is continuous), the series (1.6)is continuous.

A function that is not continuous at some point is said to be discontinuous at that point. For example, the Heaviside function (1.2)is not continuous at . In order to determine if a function is continuous at a point x in an interval ( a , b ) we apply the test:

There are two (simple discontinuity) main categories of discontinuous functions:

First kind: and exists. Then either we have or .

Second kind: a discontinuity that is not of the first kind.

Examples are:

You can check that this latter function has a discontinuity of the first kind at .

The derivative of a function is one of its fundamental properties. It represents the rate of change of the slope of the function: in other words, how fast the function changes with respect to changes in the independent variable. We focus on real-valued functions of a real variable.

Let . Then the derivative of f at x (if it exists) is defined by the limit for :

(1.7)

This limit may not exist at certain points, and it is possible to define right-hand and left-hand limits, that is, one-sided derivatives.

Some results that we learn in high school are:

(1.8)

A composite function is a function that we can differentiate using the chain rule that we state as follows:

(1.9)

A simple example of use is:

More challenging examples of composite functions are:

Taylor's theorem allows us to expand a function as a series involving higher-order derivatives of a function. We take the Cauchy form (with exact remainder):

(1.10)

and:

We conclude with a discussion of the exponential function . It is the only function that is the same as its derivative. To see this, we use the formal definition (1.7)of a derivative (and noting that ):

We summarise some useful properties of the exponential function:

For many applications we need a definition of the asymptotic behaviour of quantities such as functions and series; in particular we wish to find bounds on mathematical expressions and applications in computer science. To this end, we introduce the Landau symbols O and o.

An example is:

An example is:

We note that complexity analysis applies to both continuous and discrete functions.

In general, we are interested in functions of two (or more) variables. We consider a function of the form:

The variables x and y can take values in a given bounded or unbounded interval. First, we say that f ( x , y ) is continuous at ( a , b ) if the limit: