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

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}.