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We prove the existence of nowhere continuous bijections that satisfy theCostas property, as well as (countably and uncountably) infinite Golomb rulers.We define and prove the existence of real and rational Costas clouds, namelynowhere continuous Costas injections whose graphs are everywhere dense in aregion of the real plane, based on nonlinear solutions of Cauchy's functionalequation. We also give 2 constructive examples of a nowhere continuousfunction, that satisfies a constrained form of the Costas property (overrational or algebraic displacements only, that is), based on the indicatorfunction of a dense subset of the reals.

On nowhere continuous Costas functions and infinite Golomb rulers