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Almost continuity of the Cesàro-Vietoris function

Author: Jack B. Brown
Journal: Proc. Amer. Math. Soc. 49 (1975), 185-188
MSC: Primary 26A09
MathSciNet review: 0360943
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Abstract: Consider the following function due to Cesàro: $ \phi (0) = 0$, and if $ 0 < x \leq 1$,

$\displaystyle \phi (x) = \lim \sup ({a_1} + {a_2} + \cdots + {a_n})/n,$

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.

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Keywords: Almost continuous, connected graph, Cesàro-Vietoris function
Article copyright: © Copyright 1975 American Mathematical Society

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