IR-TST 2020 Problem 4

Given a function $g:[0,1] \to \mathbb{R}$ satisfying the property that for every non empty dissection of the trivial $[0,1]$ to subsets $A,B$ we have either $\exists x \in A; g(x) \in B$ or $\exists x \in B; g(x) \in A$ and we have furthermore $g(x)>x$ for $x \in [0,1]$ . Prove that there exist infinite $x \in [0,1]$ with $g(x)=1$ .

Proposed by Ali Zamani