We analyse some aspects of the third law of thermodynamics. We first review both the entropic version (N) and the unattainability version (U) and the relation occurring between them. Then, we heuristically interpret (N) as a continuity boundary condition for thermodynamics at the boundary T = 0 of the thermodynamic domain. On a rigorous mathematical footing, we discuss the third law both in Carathéodory's approach and in Gibbs' one. Carathéodory's approach is fundamental in order to understand the nature of the surface T = 0. In fact, in this approach, under suitable mathematical conditions, T = 0 appears as a leaf of the foliation of the thermodynamic manifold associated with the non-singular integrable Pfaffian form δQ_rev. Being a leaf, it cannot intersect any other leaf S = const of the foliation. We show that (N) is equivalent to the requirement that T = 0 is a leaf. In Gibbs' approach, the peculiar nature of T = 0 appears to be less evident because the existence of the entropy is a postulate; nevertheless, it is still possible to conclude that the lowest value of the entropy S has to be attained at the boundary of the convex set where S is defined.