Didactic transposition of structuralist praxeologies at the interface between analysis and topology: a case study

Mathematical reasoning based on structures is important for the transition from beginner to more advanced mathematics: it reflects the epistemology of the body of mathematical knowledge in more advanced mathematics courses that progresses from local reasoning about particular instances of classes of objects to more global reasoning about underlying mathematical structures through historically endorsed abstraction. We examine this type of argumentation in the sense of "structuralist praxeologies" at the interface between analysis and topology, taking as an example the equivalence of the continuity of a real-valued function on the real line with the path-connectedness of its graph. In particular, we identify phenomena of external didactic transposition in the context of options for reasonings and their implementation.