The persistence diagram is an increasingly useful tool from Topological Data Analysis, but its use alongside typical machine learning techniques requires mathematical finesse. The most success to date has come from methods that map persistence diagrams into vector spaces, in a way which maximizes the structure preserved. This process is commonly referred to as featurization. In this paper, we describe a mathematical framework for featurization called \emph{template functions}, and we show that it addresses the problem of approximating continuous functions on compact subsets of the space of persistence diagrams. Specifically, we begin by characterizing relative compactness with respect to the bottleneck distance, and then provide explicit theoretical methods for constructing compact-open dense subsets of continuous functions on persistence diagrams. These dense subsets -- obtained via template functions -- are leveraged for supervised learning tasks with persistence diagrams. Specifically, we test the method for classification and regression algorithms on several examples including shape data and dynamical systems.