Continuous digitization in Khalimsky spaces

A real-valued function defined on Rn can sometimes be approximated by a Khalimsky-continuous mapping defined on Zn. We elucidate when this can be done and give a construction for the approximation. This approximation can be used to define digital Khalimsky hyperplanes that are topological embeddings of Zn into Zn+1. In particular, we consider Khalimsky planes in Z3 and show that the intersection of two non-parallel Khalimsky planes contains a curve homeomorphic to the Khalimsky line.