This dissertation proposes an objective function based clustering approach using Lipschitzfunctions to represent the clustering function. We establish some mathematical properties including two optimality conditions and a uniqueness result; some statistical properties including two consistency results; and some computational development. This work is a step forward building upon existing work about Lipschitz classifiers to proceed from classification to clustering, also covering more theoretical and computational aspects. The mathematical contents strongly suggest further future analysis of the method. The general objective function might be of independent interest.