‘Continuous’ functions on digital pictures

Abstract

The epsilon-delta definition of continuity has a natural analog for functions that take lattice points into lattice points. It turns out that a function f is ‘continuous’ if and only if it takes neighbors into neighbors, i.e., if Q is a neighbor of P, then f( Q) = f( P) or is a neighbor of f( P) (diagonal neighbors not allowed). Some basic properties of such ‘continuous’ functions are established. In particular, we show that a ‘continuous’ function from a finite block of lattice points into itself has an ‘almost-fixed’ point P such that f( P) is a neighbor of P (diagonal neighbors allowed). We also show that a function is one-to-one and continuous if and only if it is a combination of translations, rotations by multiples of 90°, or reflections in a horizontal, vertical, or diagonal line.