Almost continuity of the Cesàro-Vietoris function

Abstract:

Consider the following function due to Cesàro: $\phi (0) = 0$, and if $0 < x \leq 1$, \[ \phi (x) = \lim \sup ({a_1} + {a_2} + \cdots + {a_n})/n,\] where the ${a_i}$ are given by the unique nonterminating binary expansion of $x = (0.{a_1}{a_2} \cdots )$. Vietoris proved in 1921 that $\phi$ is connected (as a subset of $[0,1] \times R$). The purpose of this note is to alter Vietoris’s argument in order to prove that $\phi$ is actually almost continuous in the sense of Stallings, thus answering a question raised recently by B. D. Smith.