6.8. Continuously Extendable FunctionsWith a function f continuous at a we can calculate the value $f(a)$ knowing alone the values of f in a neighbourhood of a. When doing this there is actually no need for $f(a)$ to exist. The mere calculation works even if a does not belong to the domain of f. This observation will enable us to assign additional values to certain functions!
Consider:
As there are sufficient many values of f in any neighbourhood of a we can prove the uniqueness of a continuous extension.
Consider:
In a first series of examples we consider polynomial quotients. The crucial points are the denominator's zeros. The following criterion is a valuable tool when studying these kind of functions.
Some examples will practise the zero criterion:
With more general functions it is often rather difficult to check their limit behaviour. The following criterion at least offers a technical approach for an investigation.
The following examples illustrate the sequence criterion. The first one also proves that the zero criterion [6.8.3] may fail with arbitrary quotients.
To extend a function continuously is a concept primarily designed to assign additional values to certain functions. But it also contributes to a further problem: Which are the conditions that allow to continuously concatenate two functions at a given point?
Consider:
The ability of two functions to be continuously concatenable is solely controlled by their limit behaviour at a.
