Costas permutations in the continuum

We extend the definition of the Costas property to functions in the continuum, namely on intervals of the reals or the rationals, and argue that such functions can be used in the same applications as discrete Costas arrays. We construct Costas bijections in the real continuum within the class of piecewise continuously differentiable functions; over the rationals we propose a non-smooth construction of great generality and flexibility whose success, though, relies heavily on their enumerability, and therefore cannot be generalized over the reals in an obvious way.