  We show that every isoperimetric set in R^N with density is bounded if the
density is continuous and bounded by above and below. This improves the
previously known boundedness results, which basically needed a Lipschitz
assumption; on the other hand, the present assumption is sharp, as we show with
an explicit example. To obtain our result, we observe that the main tool which
is often used, namely a classical "\epsilon-\epsilon" property already
discussed by Allard, Almgren and Bombieri, admits a weaker counterpart which is
still sufficient for the boundedness, namely, an "\epsilon-\epsilon^\beta"
version of the property. And in turn, while for the validity of the first
property the Lipschitz assumption is essential, for the latter the sole
continuity is enough. We conclude by deriving some consequences of our result
about the existence and regularity of isoperimetric sets.
