A characterization of Beckenbach families admitting discontinuous Jensen affine functions (Q2915419)

A set of continuous functions defined on a real interval \(I\) is called a Beckenbach family if arbitrary two points of \(I\times\mathbb R\) (with distinct first coordinates) can be interpolated by a unique member of the family. We say that \(\phi\colon I\rightarrow\mathbb R\) is Jensen affine (with respect to the underlying family), if NEWLINE\[NEWLINE \phi\left(\frac{x+y}{2}\right) =\psi\left(\frac{x+y}{2}\right) NEWLINE\]NEWLINE holds where \(\psi\) denotes the (unique) member of the family fulfilling \(\phi(x)=\psi(x)\) and \(\phi(y)=\psi(y)\). The classical example for a Beckenbach family is the set of usual affine functions, that is, functions of the form \(ax+b\).NEWLINENEWLINE It is well known that, in the classical setting, there exist discontinuous Jensen affine functions. Matkowski characterized those Beckenbach families among functions of the form \(a\alpha(x)+b\) (where \(\alpha\) is continuous and strictly monotonic) which consist of discontinuous affine functions. Motivated by this result, the author gives a description of Beckenbach families defined on the whole real line containing monotonic functions which consist of discontinuous Jensen affine functions.