Sierpinski-Zygmund functions that have the Cantor intermediate value property

by

Krzysztof Banaszewski and Tomasz Natkaniec

7 pages

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 2c 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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