Numerical Methods in Computational Finance. Daniel J. Duffy

      1.2.2 An Example

It can be a mathematical challenge to prove that a function is continuous using the above ‘epsilon-delta’ approach in Definition 1.1. One approach is to use the well-known technique of splitting the problem into several mutually exclusive cases, solving each case separately and then merging the corresponding partial solutions to form the desired solution. To this end, let us examine the square root function:

      (1.3)

      We show that there exists

      Then:

      We now consider two cases:

       Case 1 : . Then:Choose .

       

       Case 2: . Then:Hence:Choose .

      We have thus proved that the square root function is continuous.

      1.2.3 Uniform Continuity

In general terms, uniform continuity guarantees that f (x) and f (y) can be made as close to each other as we please by requiring that x and y be sufficiently close to each other. This is in contrast to ordinary continuity, where the distance between f (x) and f (y) 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

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