Question 75
Given a function $g : [0, 1] \to \mathbb{R}$ such that for every non-empty partition of the interval $[0, 1]$ into subsets $A, B$ either $\exists x \in A : g(x) \in B$ or $\exists x \in B : g(x) \in A$. Furthermore, $g(x) \gt x$ for all $x \in [0, 1]$. Prove that $g(x) = 1$, for infinitely many $x$ in its domain.