Limit-like predictability for discontinuous functions

Abstract:

Our starting point is the following question: To what extent is a function’s value at a point $x$ of a topological space determined by its values in an arbitrarily small (deleted) neighborhood of $x$? For continuous functions, the answer is typically “always” and the method of prediction of $f(x)$ is just the limit operator. We generalize this to the case of an arbitrary function mapping a topological space to an arbitrary set. We show that the best one can ever hope to do is to predict correctly except on a scattered set. Moreover, we give a predictor whose error set, in $T_0$ spaces, is always scattered.