6.9. Properties of Continuously Extendable FunctionsIt is quite natural that continuously extendable functions inherit a lot of their properties from the continuous ones. But there are also some results from sequences  the nesting theorem and the limit theorems to name a few  that have mirror theorems within our current subject. As with continuity itself continuous extendability proves to be a local property. The initial propositions work out the details.
The local character is also stressed by the fact that two functions that coincide locally don't differ in their limit behaviour (cf. [6.2.11]).
Consider:
Similar to [6.4.1] and [6.4.9] we can notice the typical interaction between fixed value of f and its neighbouring values. Again we only note the result for $</\le$.
When studying limits of sequences the limit theorems proved to be a valuable tool. Fortunately they are still at hand for limits of functions.
And we have a fifth limit theorem as the composition of functions is also compatible with the continuous extendability.
Alongside with tools for calculating limits we need methods for estimating limits.
The nesting theorem is often used after succeeding in estimating a difference by absolute value. The following proposition guarantees that this practice leads to limits.
