MathDB

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

\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

Problem Statement