We construct (in ZFC) an example of Sierpinski-Zygmund
function having the Cantor intermediate value property CIVP and
observe that every such function does not have the strong Cantor
intermediate value property SCIVP, which solves a problem of
R. Gibson. Moreover we prove that both
families: SCIVP functions and CIVP\setminus SCIVP functions
are 2^{c} dense in the uniform closure of the class of
CIVP functions. We show also that if the real line
is not a union of less than continuum many its meager subsets,
then there exists an almost continuous Sierpinski-Zygmund
function having the Cantor intermediate value property. Because
such a function does not have the strong Cantor intermediate
value property, it is not extendable. This solves another
problem of Gibson.

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**Last modified August 9, 1997.**