  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, but our attempts to construct
a fractal-like Costas bijection there are successful only under slight but
necessary deviations from the usual arithmetic laws. Furthermore, we are able,
contingent on the validity of Artin's conjecture, to set up a limiting process
according to which sequences of Welch Costas arrays converge to smooth Costas
bijections over the reals. The situation over the rationals is different:
there, we propose an algorithm of great generality and flexibility for the
construction of a Costas fractal bijection. Its success, though, relies heavily
on the enumerability of the rationals, and therefore it cannot be generalized
over the reals in an obvious way.
