MATHEMATICS

Continuity of convex functions

A. V. Polukhina, T. E. Khmyleva

Tomsk State University

Abstract: In this paper, we consider the set $V(K)$ of all convex real-valued functions defined on convex compacts $K\subset\mathbb R^n$ and find conditions under which all functions $f\in V(K)$ are scattered continuous. It is shown that there exist functions $f\in V(K)$ that are not Borel, and, for any ordinal $\alpha<\omega_1$, there are functions $f\in V(K)$ that exactly belong to the $\alpha$th Baire class.

Keywords: convex function, scattered continuous functions, extreme points, Borel sets, ordinals, compact.

UDC: 515.12

Received: 25.07.2013