This paper proposes a new mathematical model for the open pit mine planning problem, based on continuous functional analysis. The traditional models for this problem have been constructed by using discrete 0-1 decision variables, giving rise to large-scale combinatorial and Mixed Integer Programming (MIP) problems. Instead, we use a continuous approach which allows for a refined imposition of slope constraints associated with geotechnical stability. The model introduced here is posed in a suitable functional space, essentially the real-valued functions that are Lipschitz continuous on a given two dimensional bounded region. We derive existence results and investigate some qualitative properties of the solutions.