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

The $\epsilon-\epsilon^\beta$ property, the boundedness of isoperimetric sets in $\R^N$ with density, and some applications