I recently discovered a neat unique mapping from an integer $0 \leq j < N^2$ to a point $(x(j),y(j))$, using $x(j) = (j \mod N)$ and $y(j) = \left\lfloor{\frac{j}{N}}\right\rfloor$, and wondered if there was some similar function that maps the continuous interval (0,1) onto the unit square.
Of course, linear algebra has taught me that no continuous such mapping exists, but is there a definable discontinuous function to search for? Each set of points has the same number of elements after all..?