Numerical Methods in Computational Finance. Daniel J. Duffy

may depend on x and y themselves. In other words, in Definition 1.1

depends only on

and not on the points in the domain. Continuity itself is a local property because a function f is or is not continuous at a particular point and continuity can be determined by looking at the values of the function in an arbitrary small neighbourhood of that point. Uniform continuity, on the other hand, is a global property of f because the definition refers to pairs of points rather than individual points. The new definition in this case for a function f defined in an interval I is:

      Let us take an example of a uniformly continuous function:

      (1.4)

      Then

      Choose

.

      In general, a continuous function on a closed interval is uniformly continuous. An example is:

      (1.5)

      Let

. Then:

      Choose

.

      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.

      1.2.4 Classes of Discontinuous Functions

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

.

1.3 DIFFERENTIAL CALCULUS

      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

: