If I were good at making animated gifs I'd have fun tweeting about pointwise convergence, uniform convergence, L^1 convergence and so on, showing sequences of functions that converge in one way but not another. Analysis to me is quite visual: I understand it with mental pictures.
A small point: Barry Mazur has publicly attested to the pivotal role in his own early development played by the long line. So perhaps it is more useful than it seems at first sight.