Set of Points of Continuity and Maximally Discontinuous Extensions

The Baire category theorem has numerous applications in real and functional analysis. In this note, we discuss an interesting application in the context of metric spaces. We characterize all subsets of a general metric space that appear as the set of continuity points for functions on that space. Furthermore, we juxtapose our results with the Tietze extension theorem, which beautifully leads us in another direction towards the study of extensions of continuous functions. Most of the results we discuss here already exist scattered in the literature. In this note, we strive to present them in a coherent form.