  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.
