On nowhere continuous Costas functions and infinite Golomb rulers

We prove the existence of nowhere continuous bijections that satisfy the Costas property, as well as (countably and uncountably) infinite Golomb rulers. We define and prove the existence of real and rational Costas clouds, namely nowhere continuous Costas injections whose graphs are everywhere dense in a region of the real plane, based on nonlinear solutions of Cauchy’s functional equation. We also give 2 constructive examples of a nowhere continuous function, that satisfies a constrained form of the Costas property (over rational or algebraic displacements only, that is), based on the indicator function of a dense subset of the reals.