Question

A function to a discrete topology is continuous only it is the constant function

Answer 1

If XX is equipped with discrete topology then every function f:X→Zf:X→Z (where ZZ denotes an arbitrary topological space) is continuous.

This because the preimage of every subset of ZZ w.r.t. ff is an open set.

Note that these preimages are subsets of XX and every subset of XX is open. This because XX is equipped with the discrete topology: τX=℘(X)τX=℘(X).