Barron-type estimators are histogram-based distribution estimators that have been proved to have good consistency properties according to several information theoretic criteria. However they are not continuous. In this paper, we examine a new class of continuous distribution estimators obtained as a combination of Barron-type estimators with the frequency polygon. We prove the consistency of these estimators in expected information divergence and expected chi(2)-divergence. For one of them we evaluate the rate of convergence in expected chi(2)-divergence.