Numerical Methods in Computational Finance. Daniel J. Duffy
Читать онлайн книгу.
may depend on x and y themselves. In other words, in Definition 1.1
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
Choose
An example of a function that is continuous and nowhere differentiable is the Weierstrass function that we can write as a Fourier series:
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
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
