Finding The Sign Of A Function Value By Binary Cellular Automaton

Given a continuous function f(x), suppose that the sign of f has only finitely many discontinuous points in the interval [0, 1]. We show how to use a sequence of one-dimensional deterministic binary cellular automata to determine the sign of f(ρ) where ρ is the (number) density of 1s in an arbitrarily given bit string of finite length provided that f satisfies certain technical conditions.