Читать книгу Numerical Methods in Computational Finance - Daniel J. Duffy - Страница 18

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.