I studied a bit of chaos in college, but from my learning, I've never known a mapping that does not consist of discrete orbits. For example, take the Mandelbrot set, where the points on the mapping are simply iterations of the function: $f(x), f^2(x), f^3(x), ...$
And so my conclusion has always been that a chaotic mapping is necessarily both discontinuous and countable.
But is that true?