This paper examines the distributional properties of poverty measures which are discontinuous at the poverty line. It is shown that among all the additive poverty measures, only those measures with some discontinuous jump at the poverty line are such that it is optimal to allocate a given antipoverty budget either to the richest of the poor, or to the poorest of the poor, or to both. A special class of such poverty measures is an extension of the well-known 𝑃𝛼, the properties of which are investigated.