Peano curves
Show that there is a continuous function f from the unit interval [0, 1] onto the unit square [0, 1] × [0, 1]. Hints: Let f be the limit of a sequence of functions fn which will be piecewise linear. Let f1(t ) ≡ (0, t ). Let f2(t ) = (2t, 0) for 0 ≤ t ≤ 1/4, f2(t ) = (1/2, 2t - 1/2) for 1/4 ≤ t ≤ 3/4, and f2(t ) = (2 - 2t, 1) for 3/4 ≤ t ≤ 1. At the nth stage, the unit square is divided into 2n · 2n = 4n equal squares, where the graph of fn runs along at least one edge of each of the small squares. Then at the next stage, on the interval where fn ran along one such edge, fn+1 will first go halfway along a perpendicular edge, then along a bisector parallel to the original edge, then back to the final vertex, just as f2 related to f1. Show that this scheme can be carried through, with fn converging uniformly to f .