  We study the decreasing rearrangement of functions in VMO, and show that for
rearrangeable functions, the mapping f -> f* preserves vanishing mean
oscillation. Moreover, as a map on BMO, while bounded, it is not continuous,
but continuity holds at points in VMO (under certain conditions). This also
applies to the symmetric decreasing rearrangement. Many examples are included
to illustrate the results.
