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Let be a continuum. Define
by
. It is known that for any continuum the function is upper
semi-continuous, [Nadler 1978, Theorem 15.2, p. 514], and it is continuous on
a dense subset of , [Nadler 1978, Corollary 15.3, p. 515].
A continuum is said to be *-smooth at*
provided that
the function is continuous at . A continuum is said to be
*-smooth* provided that the function is continuous on
, i.e., at each
(see [Nadler 1978, Definition 5.15, p. 517]).

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*Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih*

*2001-11-30*