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This paper catalogues a variety of examples concerning a type of function ofa $p$-adic integer variable defined by a formal series expression we havedubbed "$\mathcal{F}$-series". These series exhibit a new, previouslyundocumented form of point-wise convergence, one where the topology in whichthe limit of a sequence of functions $\left\{ f_{n}\right\}_{n\geq1}$ convergesdepends on the point at which the sequence is evaluated. In a manner comparableto the adele ring of a number field, functions defined by $\mathcal{F}$-seriesrequire considering different metric completions of an underlying field inorder to be properly understood.

Infinite Series Whose Topology of Convergence Varies From Point to Point