Appendix: Examples of Limits from Within

In this appendix we will give a few illustrative examples. We will explore the possibility of recovering values for the functions associated to either weakly convergent or flatly divergent Fourier series, using limits from the interior of the open unit disk. In some of these simple examples we will recover the original functions in closed form. As we will see, sometimes it is also possible to write the FC functions in closed form. At the same time, we will also explore the extension of the structure to singular objects such as the Dirac delta ``function''.