On the strong continuity of convex functions

Abstract

A convex function defined on an open convex set is known to be continuous at every point of this set. In actuality, a convex function has a strengthened continuity property. In this paper, we introduce the notion of strong continuity and demonstrate that a convex function possesses this property. The proof is based only on the definition of convexity and the Jensen’s inequality. A distinct constant (constant of strong continuity) is included in the definition of strong continuity. In the article, we give an unimprovable value for this constant in the case of convex functions. The constant of strong continuity depends, in particular, on the form of the norm introduced in the space of the arguments of a convex function. Polyhedral norm is of particular interest. With its use the constant of strong continuity can be easily calculated. This requires a finite number of values of the convex function.