Here is a really broad question. What’s the lay of the land regarding maps of continuous surfaces to discrete ones? I’m thinking here of credit scores. Someone’s credit score is continuous, one-dimensional – and derives from a multitude of measures that are both continuous and discrete. As a lender, you have to decide at some point, whether or not to lend to this person – a yes/no proposition. Granted, you can charge different interest rates. So maybe it’s not a continuous-to-discrete problem? Nevertheless it has that flavour.
Logistic maps come to mind. As does the expansion of a point into several branches. Maybe someone can lay this out better…. Takers?
Added, 2014: I guess my continuous formula for the median makes a bridge between continuous & discrete.
Added, 2015: Terry Tao: Ultraproducts as a bridge between continuous and discrete analysis (published at the end of 2013 but google eg “Furstenberg correspondence theorem”; people have been talking about some elements of his bridge for a while.
