An example which is not quite what is being asked, and which is certainly much less sophisticated than the other answers, but which comes up in my calculus teaching around this week each year: continuity and the intermediate value theorem. High school students (at least in my country) are generally not given a limit definition of continuity; ``the graph doesn't jump'', i.e. the conclusion of the IVT, is their definition of continuity. The history of this conception of continuity, in relation to the epsilon-delta/preimage-of-open-set definitions, is explored by Barany in a Notices article. Every year, one of my most interesting pedagogical challenges is to help students internalize the limit definition of continuity by seeing how it makes the IVT less obvious.

An example which is not quite what is being asked, and which is certainly much less sophisticated than the other answers, but which comes up in my calculus teaching around this week each year: continuity and the intermediate value theorem. High school students (at least in my country) are generally not given a limit definition of continuity; “the graph doesn't jump”, i.e. the conclusion of the IVT, is their definition of continuity. The history of this conception of continuity, in relation to the epsilon–delta/preimage-of-open-set definitions, is explored by Barany in a Notices article Stuck in the middle. Every year, one of my most interesting pedagogical challenges is to help students internalize the limit definition of continuity by seeing how it makes the IVT less obvious.