  This paper presents a general study of one-dimensional differentiability for
functionals on convex domains that are not necessarily open. The local
approximation is carried out by affine functionals, rather than linear ones as
in standard Gateaux differentiability. This affine notion of differentiability
naturally arises in many applications and, here and there, it appeared in the
literature. Our systematic analysis aims to give a general perspective on it.
