Daniel J. Duffy - Numerical Methods in Computational Finance

(1.1)

We can see that these functions are continuous just by drawing them. The first function is ‘smoother’ than the second function, the latter being similar to a one-factor call or put payoff on the one hand and a Rectified Linear Unit (ReLU) activation function on the other hand (Goodfellow, Bengio and Courville (2016)). Intuitively, a function f is continuous if when , no matter how x approaches p . Alternatively, small changes in x lead to small changes in .

If we formally differentiate the above ReLU function (1.1), we get the famous discontinuous Heaviside function :

(1.2)

A discontinuous function is one that is not continuous. Another discontinuous function is:

Define ; let (integer).

Then taking left and right limits gives different answers, showing that the function is not continuous.

1

2

Thus .

The following definition is based on the fact that small changes in x lead to small changes in f ( x ).

Some properties of continuous functions f(x) and g(x) are:

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 such that for :

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.

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 .