A theorem of Piccard's type and its applications to polynomial functions and convex functions of higher orders

TL;DR

This paper proves a Piccard's type theorem and demonstrates the continuity of certain polynomial and convex functions that are measurable, extending classical results to higher-order functions.

Contribution

It introduces a new Piccard's type theorem and applies it to establish the continuity of higher-order polynomial and convex functions that are measurable.

Findings

Proved a Piccard's type theorem for higher-order functions.

Established continuity of $ ext{D}$-measurable polynomial functions of order n.

Established continuity of $ ext{D}$-measurable n-convex functions.

Abstract

In the paper a theorem of Piccard's type is proved and, consequently, the continuity of $\mathcal{D}$-measurable polynomial functions of $n$-th order as well as $\mathcal{D}$-measurable $n$-convex functions is shown. The paper refers to the papers \cite{Gajda} and \cite{EJ}.