Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 21F / c / Solution

Apply part b separately to obtain continuous $g,h:X\to [0,1]$ withBecause $A\cap B=\varnothing$, the two functions never vanish simultaneously. Henceis continuous and takes values in $[0,1]$. It equals zero exactly where $g=0$, namely on $A$, and equals one exactly where $h=0$, namely on $B$.