Suppose $I=A\cup B$, where $A$ and $B$ are disjoint closed sets with $-1\in A$ and $1\in B$. Since each is the complement of the other, both are also open in $I$. Define $f=-1$ on $A$ and $f=1$ on $B$. This <locally constant function> is continuous and contradicts part (c). Therefore $A\cap B\ne\varnothing$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 11H / f / Solution

 Solution (source code)