Numerical Methods in Computational Finance. Daniel J. Duffy

      and this limit is not a rational number. The Fibonacci numbers are useful in many kinds of applications, such as optimisation (finding the minimum or maximum of a function) and random number generation.

      We define a complete metric space X as one in which every Cauchy sequence converges to an element in X. Examples of complete metric spaces are:

      Definition 1.6 An open cover of a set E in a metric space X is a collection of open subsets of X such that .

      Finally, we say that a subset K of a metric space X is compact if every open cover of K contains a finite subcover, that is for some finite N.

      1.6.3 Lipschitz Continuous Functions

      We now examine functions that map one metric space into another one. In particular, we discuss the concepts of continuity and Lipschitz continuity.

      It is convenient to discuss these concepts in the context of metric spaces.

      Definition 1.7 Let and be two metric spaces. A function f from X into Y is said to be continuous at the point if for each there exists a such that:

This is a generalisation of the concept of continuity in Section 1.2 (Definition 1.1). We should note that this definition refers to the continuity of a function at a single point. Thus, a function can be continuous at some points and discontinuous at other points.

      Definition 1.8 A function f from a metric space into a metric space is said to be a uniformly continuous on a set if for each there exists a such that:

      If the function f is uniformly continuous, then it is continuous, but the converse is not necessarily true. Uniform continuity holds for all points in the set E, whereas normal continuity is only defined at a single point.

      Definition 1.9 Let be a real-valued function and suppose that we can find two constants M and such that . Then we say that f satisfies a Lipschitz condition of order , and we write .

      We

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