Extension of the variance function of a steep exponential family

Let F={P(m,F); m[set membership, variant]MF} be a multidimensional steep natural exponential family parameterized by its domain of the means MF and let VF(m) be its variance function. This paper studies the boundary behaviour of VF. Necessary and sufficient conditions on a point of [not partial differential]MF are given so that VF admits a continuous extension to the point . It is also shown that the existence of implies the existence of a limit distribution concentrated on an exposed face of containing . The relation between and is established and some illustrating examples are given.