'A function is continuous if and only if you can draw its graph without lifting the pen from the paper' – Concept usage in proofs by students in a topology course

International audience; Many students enter university having learned that the graph of a continuous function is “in one piece” and “can be drawn without lifting the pen from the paper.” Rigorously, a function R -> R is continuous if and only if its graph is path-connected. In this article, I examine proofs of this fact by students in a topology course. Based on Moore (1994), concept usage of continuity and path-connectedness is analysed through recognition and building-with of the RBC-model of epistemic actions (Dreyfus & Kidron, 2014) in combination with a refinement of Oerter's (1982) contextual layers of objects. A “propositional” layer to describe relationships between objects used in proof is introduced and used to perform case studies of students' solutions.