Sierpiński--Zygmund Functions that have the Cantor Intermediate Value Property

Abstract

We construct (in ZFC) an example of Sierpiński--Zygmund function having the Cantor intermediate value property and observe that every such function does not have the strong Cantor intermediate value property, which solves the problem of R.Gibson [8, Question 2]. Moreover we prove that both families: $SCIVP$ functions and $CIVP\setminus SCIVP$ functions are $2^{\mathfrak{c}}$ dense in the uniform closure of the class of $CIVP$ functions. We show also that if the real line $\mathbb{R}$ is not a union of less than continuum many its meager subsets, then there exists an almost continuous Sierpiński--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[8, Question 3]{RG}.