Before revision:  In this paper we propose a new test for continuity of a density at a point based on the so-called g-order statistics, and study its properties under a novel asymptotic framework. The asymptotic frameworkis intended to approximate a small sample phenomenon: even though the total number n of observations may be large , the number of effective observations local to the cut-off is often small.
After revision:  In this paper we propose a new test for continuity of a density at a point based on the so-called g-order statistics, and study its properties under two complementary asymptotic frameworks. In the first asymptotic framework , the number of effective observations local to the cut-off is often small.