Many algorithms for shape analysis and shape processing rely on accurate estimates of differential information such as normals and curvature. In most settings, however, care must be taken around non-smooth areas of the shape where these quantities are not easily defined. This problem is particularly prominent with point-cloud data, which are discontinuous everywhere. In this paper we present an efficient and robust method for extracting principal curvatures, sharp features and normal directions of a piecewise smooth surface from its point cloud sampling, with theoretical guarantees. Our method is integral in nature and uses convolved covariance matrices of Voronoi cells of the point cloud which makes it provably robust in the presence of noise. We show analytically that our method recovers correct principal curvatures and principal curvature directions in smooth parts of the shape, and correct feature directions and feature angles at the sharp edges of a piecewise smooth surface, with the error bounded by the Hausdorff distance between the point cloud and the underlying surface. Using the same analysis we provide theoretical guarantees for a modification of a previously proposed normal estimation technique. We illustrate the correctness of both principal curvature information and feature extraction in the presence of varying levels of noise and sampling density on a variety of models.