A function whose domain is all reals, and which maps all rational numbers to 1 and all irrational

Question

numbers to 0.

Answer 1

not much to do except define at described. There are several functions which work with this kind of condition:

Thomae's function, which is 1/q when x=p/q (rational), and 0 when x is irrational

Dirichlet function is just as you describe